We study the fair allocation of indivisible goods with variable groups. In this model, the goal is to partition the agents into groups of given sizes and allocate the goods to the groups in a fair manner. We show that for any number of groups and corresponding sizes, there always exists an envy-free up to one good (EF1) outcome, thereby generalizing an important result from the individual setting. Our result holds for arbitrary monotonic utilities and comes with an efficient algorithm. We also prove that an EF1 outcome is guaranteed to exist even when the goods lie on a path and each group must receive a connected bundle. In addition, we consider a probabilistic model where the utilities are additive and drawn randomly from a distribution. We show that if there are n agents and the number of goods m is divisible by the number of groups k, then an envy-free outcome exists with high probability if m = ω(log n), and this bound is tight. On the other hand, if m is not divisible by k, then an envy-free outcome is unlikely to exist as long as m = o(√n).

Rank aggregation is a task of combining the rankings of items from multiple users into a single ranking that best represents the users' rankings. Alabi et al. (AAAI'22) presents differentially-private (DP) polynomial-time approximation schemes (PTASes) and 5-approximation algorithms with certain additive errors for the Kemeny rank aggregation problem in both central and local models. In this paper, we present improved DP PTASes with smaller additive error in the central model. Furthermore, we are first to study the footrule rank aggregation problem under DP. We give a near-optimal algorithm for this problem; as a corollary, this leads to 2-approximation algorithms with the same additive error as the 5-approximation algorithms of Alabi et al. for the Kemeny rank aggregation problem in both central and local models.

We study the problem of generating synthetic data under differential privacy. We establish fixed-parameter tractability (FPT) for this problem where the parameter is the treewidth of the query family’s incidence graph. Our algorithms attain optimal error rates across all regimes and are realized by two different approaches: the first is based on linear programming (LP) and the FPT of the separation problem for the LP dual; the second is based on a subsampled private multiplicative weights method, where we obtain FPT for sampling from Gibbs distributions. Both approaches are unified by a dynamic programming framework over a tree decomposition.

User-level differentially private (DP) stochastic convex optimization has garnered significant attention due to the paramount importance of safeguarding user privacy in modern large-scale ML applications. Current methods, such as those based on DP stochastic gradient descent (SGD), often struggle with high gradient computation complexity or suboptimal utility due to the need to privatize every intermediate iterate. In this work, we introduce a new nearly linear-time algorithm that resolves this trade-off and achieves the optimal excess rates via an adaptive outlier removal framework. Our key innovation is integrating the sparse vector technique directly into the SGD loop, supported by a novel robust divergence analysis. This approach naturally bounds the sensitivity of gradient estimates without requiring privatization of all intermediate steps. Specifically, our mechanism computes a local concentration score to adaptively filter out users whose updates diverge from the population geometry. Crucially, this approach preserves the unbiasedness of the gradient estimate in well-concentrated regimes while strictly bounding sensitivity in the presence of outliers. We also explore extensions to the $\ell_\infty$ setting demonstrating the generality of our analysis.

We study the $d$-dimensional knapsack problem. We are given a set of items, each with a $d$-dimensional cost vector and a profit, along with a $d$-dimensional budget vector. The goal is to select a set of items that do not exceed the budget in all dimensions and maximize the total profit. A PTAS with running time $n^{Θ(d/\varepsilon)}$ has long been known for this problem, where $\varepsilon$ is the error parameter and $n$ is the encoding size. Despite decades of active research, the best running time of a PTAS has remained $O(n^{\lceil d/\varepsilon \rceil - d})$. Unfortunately, existing lower bounds only cover the special case with two dimensions $d = 2$, and do not answer whether there is a $n^{o(d/\varepsilon)}$-time PTAS for larger values of $d$. The status of exact algorithms is similar: there is a simple $O(n \cdot W^d)$-time (exact) dynamic programming algorithm, where $W$ is the maximum budget, but there is no lower bound which explains the strong exponential dependence on $d$. In this work, we show that the running times of the best-known PTAS and exact algorithm cannot be improved up to a polylogarithmic factor assuming Gap-ETH. Our techniques are based on a robust reduction from 2-CSP, which embeds 2-CSP constraints into a desired number of dimensions, exhibiting tight trade-off between $d$ and $\varepsilon$ for most regimes of the parameters. Informally, we obtain the following main results for $d$-dimensional knapsack. No $n^{o(d/\varepsilon \cdot 1/(\log(d/\varepsilon))^2)}$-time $(1-\varepsilon)$-approximation for every $\varepsilon = O(1/\log d)$. No $(n+W)^{o(d/\log d)}$-time exact algorithm (assuming ETH). No $n^{o(\sqrt{d})}$-time $(1-\varepsilon)$-approximation for constant $\varepsilon$. $(d \cdot \log W)^{O(d^2)} + n^{O(1)}$-time $Ω(1/\sqrt{d})$-approximation and a matching $n^{O(1)}$-time lower~bound.

We consider a setting where we have a ground set ℳ together with real-valued set functions f₁, … , f_n, and the goal is to partition ℳ into two sets S₁,S₂ such that |f_i(S₁) - f_i(S₂)| is small for every i. Many results in discrepancy theory can be stated in this form with the functions f_i being additive. In this work, we initiate the study of the unstructured case where f_i is not assumed to be additive. We show that even without the additivity assumption, the upper bound remains at most O(√{n log n}). Our result has implications on the fair allocation of indivisible goods. In particular, we show that a consensus halving up to O(√{n log n}) goods always exists for n agents with monotone utilities. Previously, only an O(n) bound was known for this setting.

Abstract. When dividing items among agents, two of the most widely studied fairness notions are envy-freeness and proportionality. We consider a setting where [Formula: see text] chores are allocated to [Formula: see text] agents and the disutility of each chore for each agent is drawn from a probability distribution. We show that an envy-free allocation exists with high probability provided that [Formula: see text], and moreover, [Formula: see text] must be at least [Formula: see text] in order for the existence to hold. On the other hand, we prove that a proportional allocation is likely to exist as long as [Formula: see text], and this threshold is asymptotically tight. Our results reveal a clear contrast with the allocation of goods, where a larger number of items is necessary to ensure existence for both notions.

We introduce the \emph{Balls-and-Bins} sampling for differentially private (DP) optimization methods such as DP-SGD. While it has been common practice to use some form of shuffling in DP-SGD implementations, privacy accounting algorithms have typically assumed that Poisson subsampling is used instead. Recent work by Chua et al. (2024), however, pointed out that shuffling based DP-SGD can have a much larger privacy cost in practical regimes of parameters. In this work we show that the Balls-and-Bins sampling achieves the "best-of-both" samplers, namely, the implementation of Balls-and-Bins sampling is similar to that of Shuffling and models trained using DP-SGD with Balls-and-Bins sampling achieve utility comparable to those trained using DP-SGD with Shuffling at the same noise multiplier, and yet, Balls-and-Bins sampling enjoys similar-or-better privacy amplification as compared to Poisson subsampling in practical regimes.

We introduce $\mathsf{PREM}$ (Private Relative Error Multiplicative weight update), a new framework for generating synthetic data that achieves a {\em relative} error guarantee for statistical queries under $(\varepsilon, \delta)$-differential privacy (DP). Namely, for a domain ${\cal X}$, a family ${\cal F}$ of queries $f : {\cal X} \to \{0, 1\}$, and $\zeta > 0$, our framework yields a mechanism that on input dataset $D \in {\cal X}^n$ outputs a synthetic dataset $\widehat{D} \in {\cal X}^n$ such that all statistical queries in ${\cal F}$ on $D$, namely $\sum_{x \in D} f(x)$ for $f \in {\cal F}$, are within a $1 \pm \zeta$ {\em multiplicative} factor of the corresponding value on $\widehat{D}$ up to an {\em additive} error that is polynomial in $\log |{\cal F}|$, $\log |{\cal X}|$, $\log n$, $\log(1/\delta)$, $1/\varepsilon$, and $1/\zeta$. In contrast, any $(\varepsilon, \delta)$-DP mechanism is known to require worst-case additive error that is polynomial in at least one of $n, |{\cal F}|$, or $|{\cal X}|$. We complement our algorithm with nearly matching lower bounds.

Machine unlearning algorithms, designed for selective removal of training data from models, have emerged as a promising approach to growing privacy concerns. In this work, we expose a critical yet underexplored vulnerability in the deployment of unlearning systems: the assumption that the data requested for removal is always part of the original training set. We present a threat model where an attacker can degrade model accuracy by submitting adversarial unlearning requests for data \textit{not} present in the training set. We propose white-box and black-box attack algorithms and evaluate them through a case study on image classification tasks using the CIFAR-10 and ImageNet datasets, targeting a family of widely used unlearning methods. Our results show extremely poor test accuracy following the attack—3.6% on CIFAR-10 and 0.4% on ImageNet for white-box attacks, and 8.5% on CIFAR-10 and 1.3% on ImageNet for black-box attacks. Additionally, we evaluate various verification mechanisms to detect the legitimacy of unlearning requests and reveal the challenges in verification, as most of the mechanisms fail to detect stealthy attacks without severely impairing their ability to process valid requests. These findings underscore the urgent need for research on more robust request verification methods and unlearning protocols, should the deployment of machine unlearning systems become more prevalent in the future.

We study the allocation of indivisible goods under conflicting constraints, represented by a graph. In this framework, vertices correspond to goods and edges correspond to conflicts between a pair of goods. Each agent is allocated an independent set in the graph. In a recent work of Kumar et al. (AAMAS, 2024), it was shown that a maximal EF1 allocation exists for interval graphs and two agents with monotone valuations. We significantly extend this result by establishing that a maximal EF1 allocation exists for any graph when the two agents have monotone valuations. To compute such an allocation, we present a polynomial-time algorithm for additive valuations, as well as a pseudo-polynomial time algorithm for monotone valuations. Moreover, we complement our findings by providing a counterexample demonstrating a maximal EF1 allocation may not exist for three agents with monotone valuations; further, we establish NP-hardness of determining the existence of such allocations for every fixed number n >= 3 of agents. All of our results for goods also apply to the allocation of chores.

Several resource allocation settings involve agents with unequal entitlements represented by weights. We analyze weighted fair division from an asymptotic perspective: if m items are divided among n agents whose utilities are independently sampled from a probability distribution, when is it likely that a fair allocation exist? We show that if the ratio between the weights is bounded, a weighted envy-free allocation exists with high probability provided that m = Omega(n log n / log log n), generalizing a prior unweighted result. For weighted proportionality, we establish a sharp threshold of m = n / (1 - \mu) for the transition from non-existence to existence, where \mu in (0,1) denotes the mean of the distribution. In addition, we prove that for two agents, a weighted envy-free (and weighted proportional) allocation is likely to exist if m = omega(sqrt{r}), where r denotes the ratio between the two weights.

We study differential privacy (DP) in a multi-party setting where each party only trusts a (known) subset of the other parties with its data. Specifically, given a trust graph where vertices correspond to parties and neighbors are mutually trusting, we give a DP algorithm for aggregation with a much better privacy-utility trade-off than in the well-studied local model of DP (where each party trusts no other party). We further study a robust variant where each party trusts all but an unknown subset of at most t of its neighbors (where t is a given parameter), and give an algorithm for this setting. We complement our algorithms with lower bounds, and discuss implications of our work to other tasks in private learning and analytics.

The conventional approach in differential privacy (DP) literature formulates the privacy-utility tradeoff with a "privacy-first" perspective: for a predetermined level of privacy, a certain utility is achievable. However, practitioners often operate under a "utility-first" paradigm, prioritizing a desired level of utility and then determining the corresponding privacy cost. Wu et al. [2019] initiated a formal study of this ``utility-first'' perspective by introducing ex-post DP. They demonstrated that by adding correlated Laplace noise and progressively reducing it on demand, a sequence of increasingly accurate estimates of a private parameter can be generated, with the privacy cost attributed only to the least noisy iterate released. This led to a Laplace mechanism variant that achieves a specified utility with minimal privacy loss. However, their work, and similar findings by Whitehouse et al. [2023], are primarily limited to simple mechanisms based on Laplace or Gaussian noise. In this paper, we significantly generalize these results. In particular, we extend the findings of Wu et al. [2019] and Liu and Talwar [2019] to support any sequence of private estimators, incurring at most a doubling of the original privacy budget. Furthermore, we demonstrate that hyperparameter tuning for these estimators, including the selection of an optimal privacy budget, can be performed without additional privacy cost. Finally, we extend our results to ex-post R\'{e}nyi DP, further broadening the applicability of utility-first privacy mechanisms.

Fairness and privacy are two important concerns in social decision-making processes such as resource allocation . We initiate the study of privacy in fair division by investigating the fair allocation of indivisible resources using the well-established framework of differential privacy. We present algorithms for approximate envy-freeness and proportionality when two instances are considered to be adjacent if they differ only on the utility of a single agent for a single item. On the other hand, we provide strong negative results for both fairness criteria when the adjacency notion allows the entire utility function of a single agent to change.

We study the complexity of a fundamental algorithm for fairly allocating indivisible items, the round-robin algorithm. For n agents and m items, we show that the algorithm can be implemented in time O ( n m log ⁡ ( m / n ) ) in the worst case. If the agents' preferences are uniformly random, we establish an improved (expected) running time of O ( n m + m log ⁡ m ) . On the other hand, assuming comparison queries between items, we prove that Ω ( n m + m log ⁡ m ) queries are necessary to implement the algorithm, even when randomization is allowed. We also derive bounds in noise models where the answers to queries are incorrect with some probability. Our proofs involve novel applications of tools from multi-armed bandits, information theory, as well as posets and linear extensions.

Browser fingerprinting can be used to identify and track users across the Web, even without cookies, by collecting attributes from users' devices to create unique 'fingerprints'. This technique and resulting privacy risks have been studied for over a decade. Yet further research is limited because prior studies used data not publicly available. Additionally, data in prior studies lacked user demographics. Here we provide a first-of-its-kind dataset to enable further research. It includes browser attributes with users' demographics and survey responses, collected with informed consent from 8,400 US study participants. We use this dataset to demonstrate how fingerprinting risks differ across demographic groups. For example, we find lower income users are more at risk, and find that as users' age increases, they are both more likely to be concerned about fingerprinting and at real risk of fingerprinting. Furthermore, we demonstrate an overlooked risk: user demographics, such as gender, age, income level and race, can be inferred from browser attributes commonly used for fingerprinting, and we identify which browser attributes most contribute to this risk. Our data collection process also conducted an experiment to study what impacts users' likelihood to share browser data for open research, in order to inform future data collection efforts, with responses from 12,461 total participants. Female participants were significantly less likely to share their browser data, as were participants who were shown the browser data we asked to collect. Overall, we show the important role of user demographics in the ongoing work that intends to assess fingerprinting risks and improve user privacy, with findings to inform future privacy enhancing browser developments. The dataset and data collection tool we provide can be used to further study research questions not addressed in this work.

The Privacy Sandbox initiative from Google includes APIs for enabling privacy-preserving advertising functionalities as part of the effort to limit third-party cookies. In particular, the Private Aggregation API (PAA) and the Attribution Reporting API (ARA) can be used for ad measurement while providing different guardrails for safeguarding user privacy, including a framework for satisfying differential privacy (DP). In this work, we provide an abstract model for analyzing the privacy of these APIs and show that they satisfy a formal DP guarantee under certain assumptions. Our analysis handles the case where both the queries and database can change interactively based on previous responses from the API.

We investigate fairness in the allocation of indivisible items among groups of agents using the notion of maximin share (MMS). While previous work has shown that no nontrivial multiplicative MMS approximation can be guaranteed in this setting for general group sizes, we demonstrate that ordinal relaxations are much more useful. For example, we show that if n agents are distributed equally across g groups, there exists a 1-out-of- k MMS allocation for k = O ( g log ⁡ ( n / g ) ) , while if all but a constant number of agents are in the same group, we obtain k = O ( log ⁡ n / log ⁡ log ⁡ n ) . We also establish the tightness of these bounds and provide non-asymptotic results for the case of two groups. Our proofs leverage connections to combinatorial covering designs.

Several resource allocation settings involve agents with unequal entitlements represented by weights. We analyze weighted fair division from an asymptotic perspective: if m items are divided among n agents whose utilities are independently sampled from a probability distribution, when is it likely that a fair allocation exist? We show that if the ratio between the weights is bounded, a weighted envy-free allocation exists with high probability provided that m = Ω ( n log n / log log n ) , generalizing a prior unweighted result. For weighted proportionality, we establish a sharp threshold of m = n / ( 1 − μ ) for the transition from non-existence to existence, where μ ∈ ( 0 , 1 ) denotes the mean of the distribution. In addition, we prove that for two agents, a weighted envy-free (and weighted proportional) allocation is likely to exist if m = ω ( r ) , where r denotes the ratio between the two weights.

We study the differentially private (DP) empirical risk minimization (ERM) problem under the \emph{semi-sensitive DP} setting where only some features are sensitive. This generalizes the Label DP setting where only the label is sensitive. We give improved upper and lower bounds on the excess risk for DP-ERM. In particular, we show that the error only scales polylogarithmically in terms of the sensitive domain size, improving upon previous results that scale polynomially in the size of the sensitive domain (Ghazi et al., NeurIPS 2021).

Label differentially private (label DP) algorithms seek to preserve the privacy of the labels in a training dataset in settings where the features are known to the adversary. In this work, we study a new family of label DP training algorithms. Unlike most prior label DP algorithms that have been based on label randomization, our algorithm naturally leverages the power of the central model of DP. It interleaves gradient projection operations with private stochastic gradient descent steps in order to improve the utility of the trained model while guaranteeing the privacy of the labels. We show that such projection-based algorithms can be made practical and that they improve on the state-of-the art for label DP training in the high-privacy regime. We complement our empirical evaluation with theoretical results shedding light on the efficacy of our method through the lens of bias-variance trade-offs.

We demonstrate a substantial gap between the privacy guarantees of the Adaptive Batch Linear Queries (ABLQ) mechanism under different types of batch sampling: (i) Shuffling, and (ii) Poisson subsampling; the typical analysis of Differentially Private Stochastic Gradient Descent (DP-SGD) follows by interpreting it as a post-processing of ABLQ. While shuffling-based DP-SGD is more commonly used in practical implementations, it has not been amenable to easy privacy analysis, either analytically or even numerically. On the other hand, Poisson subsampling-based DP-SGD is challenging to scalably implement, but has a well-understood privacy analysis, with multiple open-source numerically tight privacy accountants available. This has led to a common practice of using shuffling-based DP-SGD in practice, but using the privacy analysis for the corresponding Poisson subsampling version. Our result shows that there can be a substantial gap between the privacy analysis when using the two types of batch sampling, and thus advises caution in reporting privacy parameters for DP-SGD.

In this work, we give a new technique for analyzing individualized privacy accounting via the following simple observation: if an algorithm is one-sided add-DP, then its subsampled variant satisfies two-sided DP. From this, we obtain several improved algorithms for private combinatorial optimization problems, including decomposable submodular maximization and set cover. Our error guarantees are asymptotically tight and our algorithm satisfies pure-DP while previously known algorithms (Gupta et al., 2010; Chaturvedi et al., 2021) are approximate-DP. We also show an application of our technique beyond combinatorial optimization by giving a pure-DP algorithm for the shifting heavy hitter problem in a stream; previously, only an approximate-DP algorithm was known (Kaplan et al., 2021; Cohen & Lyu, 2023).

This paper studies $k$-claw-free graphs, exploring the connection between an extremal combinatorics question and the power of a convex program in approximating the maximum-weight independent set in this graph class. For the extremal question, we consider the notion, that we call \textit{conditional $χ$-boundedness} of a graph: Given a graph $G$ that is assumed to contain an independent set of a certain (constant) size, we are interested in upper bounding the chromatic number in terms of the clique number of $G$. This question, besides being interesting on its own, has algorithmic implications (which have been relatively neglected in the literature) on the performance of SDP relaxations in estimating the value of maximum-weight independent set. For $k=3$, Chudnovsky and Seymour (JCTB 2010) prove that any $3$-claw-free graph $G$ with an independent set of size three must satisfy $χ(G) \leq 2 ω(G)$. Their result implies a factor $2$-estimation algorithm for the maximum weight independent set via an SDP relaxation (providing the first non-trivial result for maximum-weight independent set in such graphs via a convex relaxation). An obvious open question is whether a similar conditional $χ$-boundedness phenomenon holds for any $k$-claw-free graph. Our main result answers this question negatively. We further present some evidence that our construction could be useful in studying more broadly the power of convex relaxations in the context of approximating maximum weight independent set in $k$-claw free graphs. In particular, we prove a lower bound on families of convex programs that are stronger than known convex relaxations used algorithmically in this context.

Parameterized Inapproximability Hypothesis (PIH) is a central question in the field of parameterized complexity. PIH asserts that given as input a 2-CSP on $k$ variables and alphabet size $n$, it is W[1]-hard parameterized by $k$ to distinguish if the input is perfectly satisfiable or if every assignment to the input violates 1% of the constraints. An important implication of PIH is that it yields the tight parameterized inapproximability of the $k$-maxcoverage problem. In the $k$-maxcoverage problem, we are given as input a set system, a threshold $τ>0$, and a parameter $k$ and the goal is to determine if there exist $k$ sets in the input whose union is at least $τ$ fraction of the entire universe. PIH is known to imply that it is W[1]-hard parameterized by $k$ to distinguish if there are $k$ input sets whose union is at least $τ$ fraction of the universe or if the union of every $k$ input sets is not much larger than $τ\cdot (1-\frac{1}{e})$ fraction of the universe. In this work we present a gap preserving FPT reduction (in the reverse direction) from the $k$-maxcoverage problem to the aforementioned 2-CSP problem, thus showing that the assertion that approximating the $k$-maxcoverage problem to some constant factor is W[1]-hard implies PIH. In addition, we present a gap preserving FPT reduction from the $k$-median problem (in general metrics) to the $k$-maxcoverage problem, further highlighting the power of gap preserving FPT reductions over classical gap preserving polynomial time reductions.

We provide new lower bounds on the privacy guarantee of _multi-epoch_ Adaptive Batch Linear Queries (ABLQ) mechanism with _shuffled batch sampling_, demonstrating substantial gaps when compared to _Poisson subsampling_; prior analysis was limited to a single epoch. Since the privacy analysis of Differentially Private Stochastic Gradient Descent (DP-SGD) is obtained by analyzing the ABLQ mechanism, this brings into serious question the common practice of implementing Shuffling based DP-SGD, but reporting privacy parameters as if Poisson subsampling was used. To understand the impact of this gap on the utility of trained machine learning models, we introduce a novel practical approach to implement Poisson subsampling _at scale_ using massively parallel computation, and efficiently train models with the same. We provide a comparison between the utility of models trained with Poisson subsampling based DP-SGD, and the optimistic estimates of utility when using shuffling, via our new lower bounds on the privacy guarantee of ABLQ with shuffling.

Motivated by applications of large embedding models, we study differentially private (DP) optimization problems under sparsity of _individual_ gradients. We start with new near-optimal bounds for the classic mean estimation problem but with sparse data, improving upon existing algorithms particularly for the high-dimensional regime. The corresponding lower bounds are based on a novel block-diagonal construction that is combined with existing DP mean estimation lower bounds. Next, we obtain pure- and approximate-DP algorithms with almost optimal rates for stochastic convex optimization with sparse gradients; the former represents the first nearly dimension-independent rates for this problem. Furthermore, by introducing novel analyses of bias reduction in mean estimation and randomly-stopped biased SGD we obtain nearly dimension-independent rates for near-stationary points for the empirical risk in nonconvex settings under approximate-DP.

The Privacy Sandbox Attribution Reporting API has been recently deployed by Google Chrome to support the basic advertising functionality of attribution reporting (aka conversion measurement) after deprecation of third-party cookies. The API implements a collection of privacy-enhancing guardrails including contribution bounding and noise injection. It also offers flexibility for the analyst to allocate the contribution budget. In this work, we present algorithms for optimizing the allocation of the contribution budget for summary reports from the Attribution Reporting API. We develop a synthetic data model that we find to accurately capture real-world conversion data, and extensively evaluate our method on two real-world datasets and two synthetic datasets. We show that optimizing the parameters that can be set by the analyst can significantly improve the utility achieved by querying the API while satisfying the same privacy bounds.

In this work, we study ad conversion measurement, a central functionality in digital advertising, where an advertiser seeks to estimate advertiser website (or mobile app) conversions attributed to ad impressions that users have interacted with on various publisher websites (or mobile apps). Using differential privacy (DP), a notion that has gained in popularity due to its strong mathematical guarantees, we develop a formal framework for private ad conversion measurement. In particular, we define the notion of an operationally valid configuration of the attribution rule, DP adjacency relation, contribution bounding scope and enforcement point. We then provide, for the set of configurations that most commonly arises in practice, a complete characterization, which uncovers a delicate interplay between attribution and privacy.

Abstract. We correct an error in the appendix of [N. Alon et al., SIAM J. Discrete Math., 34 (2020), pp. 885–903] and prove that it is NP-hard to approximate the size of a maximum induced matching of a bipartite graph within any constant factor.

We consider the task of producing heatmaps from users' aggregated data while protecting their privacy. We give a differentially private (DP) algorithm for this task and demonstrate its advantages over previous algorithms on real-world datasets. Our core algorithmic primitive is a DP procedure that takes in a set of distributions and produces an output that is close in Earth Mover's Distance (EMD) to the average of the inputs. We prove theoretical bounds on the error of our algorithm under a certain sparsity assumption and that these are essentially optimal.

Fairness and privacy are two important concerns in social decision-making processes such as resource allocation. We study privacy in the fair allocation of indivisible resources using the well-established framework of differential privacy. We present algorithms for approximate envy-freeness and proportionality when two instances are considered to be adjacent if they differ only on the utility of a single agent for a single item. On the other hand, we provide strong negative results for both fairness criteria when the adjacency notion allows the entire utility function of a single agent to change.

We consider the learning–unlearning paradigm defined as follows. First given a dataset, the goal is to learn a good predictor, such as one minimizing a certain loss. Subsequently, given any subset of examples that wish to be unlearnt, the goal is to learn, without the knowledge of the original training dataset, a good predictor that is identical to the predictor that would have been produced when learning from scratch on the surviving examples.We propose a new ticketed model for learning–unlearning wherein the learning algorithm can send back additional information in the form of a small-sized (encrypted) “ticket” to each participating training example, in addition to retaining a small amount of “central” information for later. Subsequently, the examples that wish to be unlearnt present their tickets to the unlearning algorithm, which additionally uses the central information to return a new predictor. We provide space-efficient ticketed learning–unlearning schemes for a broad family of concept classes, including thresholds, parities, intersection-closed classes, among others.En route, we introduce the count-to-zero problem, where during unlearning, the goal is to simply know if there are any examples that survived. We give a ticketed learning–unlearning scheme for this problem that relies on the construction of Sperner families with certain properties, which might be of independent interest.

Computational differential privacy (CDP) is a natural relaxation of the standard notion of (statistical) differential privacy (SDP) proposed by Beimel, Nissim, and Omri (CRYPTO 2008) and Mironov, Pandey, Reingold, and Vadhan (CRYPTO 2009). In contrast to SDP, CDP only requires privacy guarantees to hold against computationally-bounded adversaries rather than computationally-unbounded statistical adversaries. Despite the question being raised explicitly in several works (e.g., Bun, Chen, and Vadhan, TCC 2016), it has remained tantalizingly open whether there is any task achievable with the CDP notion but not the SDP notion. Even a candidate such task is unknown. Indeed, it is even unclear what the truth could be!In this work, we give the first construction of a task achievable with the CDP notion but not the SDP notion, under the following strong but plausible cryptographic assumptions:•Non-Interactive Witness Indistinguishable Proofs,•Laconic Collision-Resistant Keyless Hash Functions,•Differing-Inputs Obfuscation for Public-Coin Samplers.In particular, we construct a task for which there exists an $\varepsilon$-CDP mechanism with $\varepsilon=O(1)$ achieving $1-o(1)$ utility, but any $(\varepsilon, \delta)$-SDP mechanism, including computationally-unbounded ones, that achieves a constant utility must use either a super-constant $\varepsilon$ or an inverse-polynomially large $\delta$.To prove this, we introduce a new approach for showing that a mechanism satisfies CDP: first we show that a mechanism is “private” against a certain class of decision tree adversaries, and then we use cryptographic constructions to “lift” this into privacy against computationally bounded adversaries. We believe this approach could be useful to devise further tasks separating CDP from SDP.

We study the problem of performing counting queries at different levels in hierarchical structures while preserving individuals' privacy. Motivated by applications, we propose a new error measure for this problem by considering a combination of multiplicative and additive approximation to the query results. We examine known mechanisms in differential privacy (DP) and prove their optimality, under this measure, in the pure-DP setting. In the approximate-DP setting, we design new algorithms achieving significant improvements over known ones.

We introduce a new mechanism for stochastic convex optimization (SCO) with user-level differential privacy guarantees. The convergence rates of this mechanism are similar to those in the prior work of Levy et al. 2021 and Narayanan et al. 2022, but with two important improvements. Our mechanism does not require any smoothness assumptions on the loss. Furthermore, our bounds are also the first where the minimum number of users needed for user-level privacy has no dependence on the dimension and only a logarithmic dependence on the desired excess error. The main idea underlying the new mechanism is to show that the optimizers of strongly convex losses have low local deletion sensitivity, along with a new output perturbation method for functions with low local deletion sensitivity, which could be of independent interest.

Differential privacy is often applied with a privacy parameter that is larger than the theory suggests is ideal; various informal justifications for tolerating large privacy parameters have been proposed. In this work, we consider partial differential privacy (DP), which allows quantifying the privacy guarantee on a per-attribute basis. In this framework, we study several basic data analysis and learning tasks, and design algorithms whose per-attribute privacy parameter is smaller that the best possible privacy parameter for the entire record of a person (i.e., all the attributes).

In this work, we study the task of estimating the numbers of distinct and $k$-occurring items in a time window under the constraint of differential privacy (DP). We consider several variants depending on whether the queries are on general time windows (between times $t_1$ and $t_2$), or are restricted to being cumulative (between times $1$ and $t_2$), and depending on whether the DP neighboring relation is event-level or the more stringent item-level. We obtain nearly tight upper and lower bounds on the errors of DP algorithms for these problems. En route, we obtain an event-level DP algorithm for estimating, at each time step, the number of distinct items seen over the last $W$ updates with error polylogarithmic in $W$; this answers an open question of Bolot et al. (ICDT 2013).

Privacy in general, and differential privacy (DP) in particular, have become important topics in data mining and machine learning. Digital advertising is a critical component of the internet and is powered by large-scale data analytics and machine learning models; privacy concerns around these are on the rise. Despite the central importance of private ad analytics and training privacy-preserving ad prediction models, there has been relatively little exposure of this subject to the broader KDD community. In the past three years, the interest in privacy and the interest in online advertising have been steadily growing in KDD. The aim of this tutorial is to provide KDD researchers with an introduction to the problems that arise in private analytics and modeling in advertising, survey recent results, and describe the main research challenges in the space.

We study the problem, where given a dataset of $n$ i.i.d. samples from an unknown distribution $P$, we seek to generate a sample from a distribution that is close to $P$ in total variation distance, under the constraint of differential privacy. We study the settings where $P$ is a multi-dimensional Gaussian distribution with different assumptions: known covariance, unknown bounded covariance, and unknown unbounded covariance. We present new differentially private sampling algorithms, and show that they achieve near-optimal sample complexity in the first two settings. Moreover, when $P$ is a product distribution on the binary hypercube, we obtain a pure-DP algorithm whereas only an approximate-DP algorithm (with slightly worse sample complexity) was previously known.

As the use of large embedding models in recommendation systems and language applications increases, concerns over user data privacy have also risen. DP-SGD, a training algorithm that combines differential privacy with stochastic gradient descent, has been the workhorse in protecting user privacy without compromising model accuracy by much. However, applying DP-SGD naively to embedding models can destroy gradient sparsity, leading to reduced training efficiency. To address this issue, we present two new algorithms, DP-FEST and DP-AdaFEST, that preserve gradient sparsity during the private training of large embedding models. Our algorithms achieve substantial reductions ($10^6 \times$) in gradient size, while maintaining comparable levels of accuracy, on benchmark real-world datasets.

Previous work on user-level differential privacy (DP) [Ghazi et al. NeurIPS 2021, Bun et al. STOC 2023] obtained generic algorithms that work for various learning tasks. However, their focus was on the *example-rich* regime, where the users have so many examples that each user could themselves solve the problem. In this work we consider the *example-scarce* regime, where each user has only a few examples, and obtain the following results: * For approximate-DP, we give a generic transformation of any item-level DP algorithm to a user-level DP algorithm. Roughly speaking, the latter gives a (multiplicative) savings of $O_{\varepsilon,\delta}(\sqrt{m})$ in terms of the number of users required for achieving the same utility, where $m$ is the number of examples per user. This algorithm, while recovering most known bounds for specific problems, also gives new bounds, e.g., for PAC learning. * For pure-DP, we present a simple technique for adapting the exponential mechanism [McSherry & Talwar, FOCS 2007] to the user-level setting. This gives new bounds for a variety of tasks, such as private PAC learning, hypothesis selection, and distribution learning. For some of these problems, we show that our bounds are near-optimal.

We study the problem of computing pairwise statistics, i.e., ones of the form $\binom{n}{2}^{-1} \sum_{i \ne j} f(x_i, x_j)$, where $x_i$ denotes the input to the $i$th user, with differential privacy (DP) in the local model. This formulation captures important metrics such as Kendall's $\tau$ coefficient, Area Under Curve, Gini's mean difference, Gini's entropy, etc. We give several novel and generic algorithms for the problem, leveraging techniques from DP algorithms for linear queries.

We propose a new family of label randomizers for training _regression_ models under the constraint of label differential privacy (DP). In particular, we leverage the trade-offs between bias and variance to construct better label randomizers depending on a privately estimated prior distribution over the labels. We demonstrate that these randomizers achieve state-of-the-art privacy-utility trade-offs on several datasets, highlighting the importance of reducing bias when training neural networks with label DP. We also provide theoretical results shedding light on the structural properties of the optimal unbiased randomizers.

We study synthetic data release for answering multiple linear queries over a set of database tables in a differentially private way. Two special cases have been considered in the literature: how to release a synthetic dataset for answering multiple linear queries over a single table, and how to release the answer for a single counting (join size) query over a set of database tables. Compared to the single-table case, the join operator makes query answering challenging, since the sensitivity (i.e., by how much an individual data record can affect the answer) could be heavily amplified by complex join relationships. We present an algorithm for the general problem, and prove a lower bound illustrating that our general algorithm achieves parameterized optimality (up to logarithmic factors) on some simple queries (e.g., two-table join queries) in the most commonly-used privacy parameter regimes. For the case of hierarchical joins, we present a data partition procedure that exploits the concept of uniformized sensitivities to further improve the utility.

We study the problem of releasing the weights of all-pairs shortest paths in a weighted undirected graph with differential privacy (DP). In this setting, the underlying graph is fixed and two graphs are neighbors if their edge weights differ by at most 1 in the ℓ1-distance. We give an algorithm with additive error Õ(n2/3/ε) in the ε-DP case and an algorithm with additive error in the (ε,δ)-DP case, where n denotes the number of vertices. This positively answers a question of Sealfon [Sea16, Sea20], who asked whether a o(n)- error algorithm exists. We also show that an additive error of Ω(n1/6) is necessary for any sufficiently small ε,δ > 0. Furthermore, we show that if the graph is promised to have reasonably bounded weights, one can improve the error further to roughly in the ε-DP case and roughly in the (ε, δ)-DP case. Previously, it was only known how to obtain Õ(n2/3/ε1/3) additive error in the ε-DP case and additive error in the (ε,δ)-DP case for bounded-weight graphs [Sea16]. Finally, we consider a relaxation where a multiplicative approximation is allowed. We show that, with a multiplicative approximation factor k, the additive error can be reduced to Õ(n1/2+O(1/k)/ε) in the ε-DP case and Õ(n1/3+O(1/k)/ε) in the (ε,δ)-DP case.

Knockout tournaments constitute a popular format for organizing sports competitions. While prior results have shown that it is often possible to manipulate a knockout tournament by fixing the bracket, these results ignore the prevalent aspect of player seeds, which can significantly constrain the chosen bracket. We show that certain structural conditions that guarantee that a player can win a knockout tournament without seeds are no longer sufficient in light of seed constraints. On the other hand, we prove that when the pairwise match outcomes are generated randomly, all players are still likely to be knockout winners under the same probability threshold with seeds as without seeds. In addition, we investigate the complexity of deciding whether a manipulation is possible when seeds are present.

Maximum bipartite matching is a fundamental algorithmic problem which can be solved in polynomial time. We consider a natural variant in which there is a separation constraint: the vertices on one side lie on a path or a grid, and two vertices that are close to each other are not allowed to be matched simultaneously. We show that the problem is hard to approximate even for paths, and provide constant-factor approximation algorithms for both paths and grids.

Justified representation (JR) is a standard notion of representation in multiwinner approval voting. Not only does a JR committee always exist, but previous work has also shown through experiments that the JR condition can typically be fulfilled by groups of fewer than k candidates, where k is the target size of the committee. In this paper, we study such groups—known as n/k-justifying groups—both theoretically and empirically. First, we show that under the impartial culture model, n/k-justifying groups of size less than k/2 are likely to exist, which implies that the number of JR committees is usually large. We then present efficient approximation algorithms that compute a small n/k-justifying group for any given instance, and a polynomial-time exact algorithm when the instance admits a tree representation. In addition, we demonstrate that small n/k-justifying groups can often be useful for obtaining a gender-balanced JR committee even though the problem is NP-hard.

In social choice theory, (Kemeny) rank aggregation is a well-studied problem where the goal is to combine rankings from multiple voters into a single ranking on the same set of items. Since rankings can reveal preferences of voters (which a voter might like to keep private), it is important to aggregate preferences in such a way to preserve privacy. In this work, we present differentially private algorithms for rank aggregation in the pure and approximate settings along with distribution-independent utility upper and lower bounds. In addition to bounds in the central model, we also present utility bounds for the local model of differential privacy.

In multiwinner approval voting, the goal is to select k-member committees based on voters' approval ballots. A well-studied concept of proportionality in this context is the justified representation (JR) axiom, which demands that no large cohesive group of voters remains unrepresented. However, the JR axiom may conflict with other desiderata, such as coverage (maximizing the number of voters who approve at least one committee member) or social welfare (maximizing the number of approvals obtained by committee members). In this work, we investigate the impact of imposing the JR axiom (as well as the more demanding EJR axiom) on social welfare and coverage. Our approach is threefold: we derive worst-case bounds on the loss of welfare/coverage that is caused by imposing JR, study the computational complexity of finding 'good' committees that provide JR (obtaining a hardness result, an approximation algorithm, and an exact algorithm for one-dimensional preferences), and examine this setting empirically on several synthetic datasets.

We study the problem of distribution-free learning of a single neuron under adversarial label noise with respect to the squared loss. For a wide range of activation functions, including ReLUs and sigmoids, we prove hardness of learning results in the Statistical Query model and under a well-studied assumption on the complexity of refuting XOR formulas. Specifically, we establish that no polynomial-time learning algorithm, even improper, can approximate the optimal loss value within any constant factor.

We consider the computation of sparse, (ε, ϑ)-differentially private~(DP) histograms in the two-server model of secure multi-party computation~(MPC), which has recently gained traction in the context of privacy-preserving measurements of aggregate user data. We introduce protocols that enable two semi-honest non-colluding servers to compute histograms over the data held by multiple users, while only learning a private view of the data. Our solution achieves the same asymptotic l∞-error of O(log(1/ϑoverε) as in the central model of DP, but without relying on a trusted curator. The server communication and computation costs of our protocol are independent of the number of histogram buckets, and are linear in the number of users, while the client cost is independent of the number of users, ε, and ϑ. Its linear dependence on the number of users lets our protocol scale well, which we confirm using microbenchmarks: for a billion users, ε = 0.5, and ϑ = 10-11, the per-user cost of our protocol is only 1.08 ms of server computation and 339 bytes of communication. In contrast, a baseline protocol using garbled circuits only allows up to 106 users, where it requires 600 KB communication per user.

We give the first polynomial time and sample (epsilon, delta)-differentially private (DP) algorithm to estimate the mean, covariance and higher moments in the presence of a constant fraction of adversarial outliers. Our algorithm succeeds for families of distributions that satisfy two well-studied properties in prior works on robust estimation: certifiable subgaussianity of directional moments and certifiable hypercontractivity of degree 2 polynomials. Our recovery guarantees hold in the “right affine-invariant norms”: Mahalanobis distance for mean, multiplicative spectral and relative Frobenius distance guarantees for covariance and injective norms for higher moments. Prior works obtained private robust algorithms for mean estimation of subgaussian distributions with bounded covariance. For covariance estimation, ours is the first efficient algorithm (even in the absence of outliers) that succeeds without any condition-number assumptions. Our algorithms arise from a new framework that provides a general blueprint for modifying convex relaxations for robust estimation to satisfy strong worst-case stability guarantees in the appropriate parameter norms whenever the algorithms produce witnesses of correctness in their run. We verify such guarantees for a modification of standard sum-of-squares (SoS) semidefinite programming relaxations for robust estimation. Our privacy guarantees are obtained by combining stability guarantees with a new “estimate dependent” noise injection mechanism in which noise scales with the eigenvalues of the estimated covariance. We believe this framework will be useful more generally in obtaining DP counterparts of robust estimators. Independently of our work, Ashtiani and Liaw [AL21] also obtained a polynomial time and sample private robust estimation algorithm for Gaussian distributions.

We study several questions related to diversifying search results. We give improved approximation algorithms in each of the following problems, together with some lower bounds. - We give a polynomial-time approximation scheme (PTAS) for a diversified search ranking problem [Bansal et al., ICALP 2010] whose objective is to minimizes the discounted cumulative gain. Our PTAS runs in time $n^{2^{O(\log(1/ε)/ε)}} \cdot m^{O(1)}$ where $n$ denotes the number of elements in the databases. Complementing this, we show that no PTAS can run in time $f(ε) \cdot (nm)^{2^{o(1/ε)}}$ assuming Gap-ETH; therefore our running time is nearly tight. Both of our bounds answer open questions of Bansal et al. - We next consider the Max-Sum Dispersion problem, whose objective is to select $k$ out of $n$ elements that maximizes the dispersion, which is defined as the sum of the pairwise distances under a given metric. We give a quasipolynomial-time approximation scheme for the problem which runs in time $n^{O_ε(\log n)}$. This improves upon previously known polynomial-time algorithms with approximate ratios 0.5 [Hassin et al., Oper. Res. Lett. 1997; Borodin et al., ACM Trans. Algorithms 2017]. Furthermore, we observe that known reductions rule out approximation schemes that run in $n^{\tilde{o}_ε(\log n)}$ time assuming ETH. - We consider a generalization of Max-Sum Dispersion called Max-Sum Diversification. In addition to the sum of pairwise distance, the objective includes another function $f$. For monotone submodular $f$, we give a quasipolynomial-time algorithm with approximation ratio arbitrarily close to $(1 - 1/e)$. This improves upon the best polynomial-time algorithm which has approximation ratio $0.5$ by Borodin et al. Furthermore, the $(1 - 1/e)$ factor is tight as achieving better-than-$(1 - 1/e)$ approximation is NP-hard [Feige, J. ACM 1998].

We introduce a new algorithm for numerical composition of privacy random variables, useful for computing the accurate differential privacy parameters for compositions of mechanisms. Our algorithm achieves a running time and memory usage of $polylog(k)$ for the task of self-composing a mechanism, from a broad class of mechanisms, $k$ times; this class, e.g., includes the sub-sampled Gaussian mechanism, that appears in the analysis of differentially private stochastic gradient descent (DP-SGD). By comparison, recent work by Gopi et al. (NeurIPS 2021) has obtained a running time of $\widetilde{O}(\sqrt{k})$ for the same task. Our approach extends to the case of composing $k$ different mechanisms in the same class, improving upon the running time and memory usage in their work from $\widetilde{O}(k^{1.5})$ to $\wtilde{O}(k)$.

Knockout tournaments constitute a popular format for organizing sports competitions. While prior results have shown that it is often possible to manipulate a knockout tournament by fixing the bracket, these results ignore the prevalent aspect of player seeds, which can significantly constrain the chosen bracket. We show that certain structural conditions that guarantee that a player can win a knockout tournament without seeds are no longer sufficient in light of seed constraints. On the other hand, we prove that when the pairwise match outcomes are generated randomly, all players are still likely to be knockout winners under the same probability threshold with seeds as without seeds. In addition, we investigate the complexity of deciding whether a manipulation is possible when seeds are present.

We study the complexity of PAC learning halfspaces in the presence of Massart noise. In this problem, we are given i.i.d. labeled examples $(\mathbf{x}, y) \in \mathbb{R}^N \times \{ \pm 1\}$, where the distribution of $\mathbf{x}$ is arbitrary and the label $y$ is a Massart corruption of $f(\mathbf{x})$, for an unknown halfspace $f: \mathbb{R}^N \to \{ \pm 1\}$, with flipping probability $\eta(\mathbf{x}) \leq \eta < 1/2$. The goal of the learner is to compute a hypothesis with small 0-1 error. Our main result is the first computational hardness result for this learning problem. Specifically, assuming the (widely believed) subexponential-time hardness of the Learning with Errors (LWE) problem, we show that no polynomial-time Massart halfspace learner can achieve error better than $\Omega(\eta)$, even if the optimal 0-1 error is small, namely $\mathrm{OPT} = 2^{-\log^{c} (N)}$ for any universal constant $c \in (0, 1)$. Prior work had provided qualitatively similar evidence of hardness in the Statistical Query model. Our computational hardness result essentially resolves the polynomial PAC learnability of Massart halfspaces, by showing that known efficient learning algorithms for the problem are nearly best possible.

We study the problem of privately computing the $\mbox{\it anonymized histogram}$ (a.k.a. $\mbox{\it unattributed histogram}$), which is defined as the histogram without item labels. Previous works have provided algorithms with $\ell_1$- and $\ell_2^2$-errors of $O_\varepsilon(\sqrt{n})$ in the central model of differential privacy (DP).In this work, we provide an algorithm with a nearly matching error guarantee of $\widetilde{O}_\varepsilon(\sqrt{n})$ in the shuffle DP and pan-private models. Our algorithm is very simple: it just post-processes the discrete Laplace-noised histogram! Using this algorithm as a subroutine, we show applications in privately estimating symmetric properties of distributions such as entropy, support coverage, and support size.

In this paper, we consider the problem of differentially private (DP) algorithms for isotonic regression. For the most general problem of isotonic regression over a partially ordered set (poset) $\mathcal{X}$ and for any Lipschitz loss function, we obtain a pure-DP algorithm that, given $n$ input points, has an expected excess empirical risk of roughly $\mathrm{width}(\mathcal{X}) \cdot \log|\mathcal{X}| / n$, where $\mathrm{width}(\mathcal{X})$ is the width of the poset. In contrast, we also obtain a near-matching lower bound of roughly $(\mathrm{width}(\mathcal{X}) + \log |\mathcal{X}|) / n$, that holds even for approximate-DP algorithms. Moreover, we show that the above bounds are essentially the best that can be obtained without utilizing any further structure of the poset.In the special case of a totally ordered set and for $\ell_1$ and $\ell_2^2$ losses, our algorithm can be implemented in near-linear running time; we also provide extensions of this algorithm to the problem of private isotonic regression with additional structural constraints on the output function.

Abstract Tournaments can be used to model a variety of practical scenarios including sports competitions and elections. A natural notion of strength of alternatives in a tournament is a generalized king: an alternative is said to be a k-king if it can reach every other alternative in the tournament via a directed path of length at most k. In this paper, we provide an almost complete characterization of the probability threshold such that all, a large number, or a small number of alternatives are k-kings with high probability in two random models. We show that, perhaps surprisingly, all changes in the threshold occur in the range of constant k, with the biggest change being between $$k=2$$ k = 2 and $$k=3$$ k = 3 . In addition, we establish an asymptotically tight bound on the probability threshold for which all alternatives are likely able to win a single-elimination tournament under some bracket.

The privacy loss distribution (PLD) provides a tight characterization of the privacy loss of a mechanism in the context of differential privacy (DP). Recent work [18–20, 24] has shown that PLD-based accounting allows for tighter (ε, δ)-DP guarantees for many popular mechanisms compared to other known methods. A key question in PLD-based accounting is how to approximate any (potentially continuous) PLD with a PLD over any specified discrete support. We present a novel approach to this problem. Our approach supports both pessimistic estimation, which overestimates the hockey-stick divergence (i.e., δ) for any value of ε, and optimistic estimation, which underestimates the hockey-stick divergence. Moreover, we show that our pessimistic estimate is the best possible among all pessimistic estimates. Experimental evaluation shows that our approach can work with much larger discretization intervals while keeping a similar error bound compared to previous approaches and yet give a better approximation than an existing method [24].

Abstract Consider the setting where multiple parties each hold a multiset of users and the task is to estimate thereach(i.e., the number of distinct users appearing across all parties) and thefrequency histogram(i.e., fraction of users appearing a given number of times across all parties). In this work we introduce a new sketch for this task, based on an exponentially distributed counting Bloom filter. We combine this sketch with a communication-efficient multi-party protocol to solve the task in the multi-worker setting. Our protocol exhibits both differential privacy and security guarantees in the honest-but-curious model and in the presence of large subsets of colluding workers; furthermore, its reach and frequency histogram estimates have a provably small error. Finally, we show the practicality of the protocol by evaluating it on internet-scale audiences.

In this paper, we study the task of aggregating user-generated trajectories in a differentially private manner. We present a new algorithm for this problem and demonstrate its effectiveness and practicality through detailed experiments on real-world data. We also show that under simple and natural assumptions, our algorithm has provable utility guarantees.

We study the allocation of indivisible goods among groups of agents using well-known fairness notions such as envy-freeness and proportionality. While these notions cannot always be satisfied, we provide several bounds on the optimal relaxations that can be guaranteed. For instance, our bounds imply that when the number of groups is constant and the n agents are divided into groups arbitrarily, there exists an allocation that is envy-free up to Θ(n) goods, and this bound is tight. Moreover, we show that while such an allocation can be found efficiently, it is NP-hard to compute an allocation that is envy-free up to o(n) goods even when a fully envy-free allocation exists. Our proofs make extensive use of tools from discrepancy theory.

In this work, we study the trade-off between differential privacy and adversarial robustness under $L_2$-perturbations in the context of learning halfspaces. We prove nearly tight bounds on the sample complexity of robust private learning of halfspaces for a large regime of parameters. A highlight of our results is that robust and private learning is harder than robust or private learning alone. We complement our theoretical analysis with experimental results on the MNIST and USPS datasets, for a learning algorithm that is both differentially private and adversarially robust.

We study closure properties for the Littlestone and threshold dimensions of binary hypothesis classes. Given classes $\mathcal{H}_1, \ldots, \mathcal{H}_k$ of binary functions with bounded Littlestone (respectively, threshold) dimension, we establish an upper bound on the Littlestone (respectively, threshold) dimension of the class defined by applying an arbitrary binary aggregation rule to $\mathcal{H}_1, \ldots, \mathcal{H}_k$. We also show that our upper bounds are nearly tight. Our upper bounds give an exponential (in $k$) improvement upon analogous bounds shown by Alon et al. (COLT 2020), thus answering an open question posed by their work.

In this work, we study the problem of answering $k$ queries with $(\epsilon, \delta)$-differential privacy, where each query has sensitivity one. We give an algorithm for this task that achieves an expected $\ell_\infty$ error bound of $O(\frac{1}{\epsilon}\sqrt{k \log \frac{1}{\delta}})$, which is known to be tight (Steinke and Ullman, 2016). A very recent work by Dagan and Kur (2020) provides a similar result, albeit via a completely different approach. One difference between our work and theirs is that our guarantee holds even when $\delta < 2^{-\Omega(k/(\log k)^8)}$ whereas theirs does not apply in this case. On the other hand, the algorithm of Dagan and Kur (2020) has a remarkable advantage that the $\ell_{\infty}$ error bound of $O(\frac{1}{\epsilon}\sqrt{k \log \frac{1}{\delta}})$ holds not only in expectation but always (i.e., with probability one) while we can only get a high probability (or expected) guarantee on the error.

We provide an approximation algorithm for k-means clustering in the \emph{one-round} (aka \emph{non-interactive}) local model of differential privacy (DP). Our algorithm achieves an approximation ratio arbitrarily close to the best \emph{non private} approximation algorithm, improving upon previously known algorithms that only guarantee large (constant) approximation ratios. Furthermore, ours is the first constant-factor approximation algorithm for k-means that requires only \emph{one} round of communication in the local DP model, positively resolving an open question of Stemmer (SODA 2020). Our algorithmic framework is quite flexible; we demonstrate this by showing that it also yields a similar near-optimal approximation algorithm in the (one-round) shuffle DP model.

The shuffle model of differential privacy has attracted attention in the literature due to it being a middle ground between the well-studied central and local models. In this work, we study the problem of summing (aggregating) real numbers or integers, a basic primitive in numerous machine learning tasks, in the shuffle model. We give a protocol achieving error arbitrarily close to that of the (Discrete) Laplace mechanism in central differential privacy, while each user only sends 1 + o(1) short messages in expectation.

Tournaments can be used to model a variety of practical scenarios including sports competitions and elections. A natural notion of strength of alternatives in a tournament is a generalized king: an alternative is said to be a k-king if it can reach every other alternative in the tournament via a directed path of length at most k. In this paper, we provide an almost complete characterization of the probability threshold such that all, a large number, or a small number of alternatives are k-kings with high probability in two random models. We show that, perhaps surprisingly, all changes in the threshold occur in the regime of constant k, with the biggest change being between k = 2 and k = 3. In addition, we establish an asymptotically tight bound on the probability threshold for which all alternatives are likely able to win a single-elimination tournament under some bracket.

We study the allocation of indivisible goods among groups of agents using well-known fairness notions such as envy-freeness and proportionality. While these notions cannot always be satisfied, we provide several bounds on the optimal relaxations that can be guaranteed. For instance, our bounds imply that when the number of groups is constant and the $n$ agents are divided into groups arbitrarily, there exists an allocation that is envy-free up to $\Theta(\sqrt{n})$ goods, and this bound is tight. Moreover, we show that while such an allocation can be found efficiently, it is NP-hard to compute an allocation that is envy-free up to $o(\sqrt{n})$ goods even when a fully envy-free allocation exists. Our proofs make extensive use of tools from discrepancy theory.

We study the setup where each of n users holds an element from a discrete set, and the goal is to count the number of distinct elements across all users, under the constraint of (ε,δ)-differentially privacy: - In the non-interactive local setting, we prove that the additive error of any protocol is Ω(n) for any constant ε and for any δ inverse polynomial in n. - In the single-message shuffle setting, we prove a lower bound of Ω̃(n) on the error for any constant ε and for some δ inverse quasi-polynomial in n. We do so by building on the moment-matching method from the literature on distribution estimation. - In the multi-message shuffle setting, we give a protocol with at most one message per user in expectation and with an error of Õ(√n) for any constant ε and for any δ inverse polynomial in n. Our protocol is also robustly shuffle private, and our error of √n matches a known lower bound for such protocols. Our proof technique relies on a new notion, that we call dominated protocols, and which can also be used to obtain the first non-trivial lower bounds against multi-message shuffle protocols for the well-studied problems of selection and learning parity. Our first lower bound for estimating the number of distinct elements provides the first ω(√n) separation between global sensitivity and error in local differential privacy, thus answering an open question of Vadhan (2017). We also provide a simple construction that gives Ω̃(n) separation between global sensitivity and error in two-party differential privacy, thereby answering an open question of McGregor et al. (2011).

We prove several hardness results for training depth-2 neural networks with the ReLU activation function; these networks are simply weighted sums (that may include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural network that minimizes the square loss with respect to a given training set. We prove that this problem is NP-hard already for a network with a single ReLU. We also prove NP-hardness for outputting a weighted sum of k ReLUs minimizing the squared error (for k > 1) even in the realizable setting (i.e., when the labels are consistent with an unknown depth-2 ReLU network). We are also able to obtain lower bounds on the running time in terms of the desired additive error ε. To obtain our lower bounds, we use the Gap Exponential Time Hypothesis (Gap-ETH) as well as a new hypothesis regarding the hardness of approximating the well known Densest κ-Subgraph problem in subexponential time (these hypotheses are used separately in proving different lower bounds). For example, we prove that under reasonable hardness assumptions, any proper learning algorithm for finding the best fitting ReLU must run in time exponential in 1/ε². Together with a previous work regarding improperly learning a ReLU [Surbhi Goel et al., 2017], this implies the first separation between proper and improper algorithms for learning a ReLU. We also study the problem of properly learning a depth-2 network of ReLUs with bounded weights giving new (worst-case) upper bounds on the running time needed to learn such networks both in the realizable and agnostic settings. Our upper bounds on the running time essentially matches our lower bounds in terms of the dependency on ε.

In a game of persuasion with evidence, a sender has private information. By presenting evidence on the information, the sender wishes to persuade a receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender's utility depends solely on whether or not the receiver takes the action. The receiver's utility depends on both the action and the sender's private information. We study three natural variations. First, we consider the problem of computing an equilibrium of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, and polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm in algorithmic economics.

We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time $n^{Ω(\log{n})}$ (so that the state-of-the-art running time of $n^{O(\log n)}$ is optimal up to a constant in the exponent). We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter $k$: Densest $k$-Subgraph, Smallest $k$-Edge Subgraph, Densest $k$-Subhypergraph, Steiner $k$-Forest, and Directed Steiner Network with $k$ terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves $o(k)$-approximation for Densest $k$-Subgraph. This inapproximability ratio improves upon the previous best $k^{o(1)}$ factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter $k$. Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, which improves the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis.

The Randomized Response (RR) algorithm is a classical technique to improve robustness in survey aggregation, and has been widely adopted in applications with differential privacy guarantees. We propose a novel algorithm, Randomized Response with Prior (RRWithPrior), which can provide more accurate results while maintaining the same level of privacy guaranteed by RR. We then apply RRWithPrior to learn neural networks with label differential privacy (LabelDP), and show that when only the label needs to be protected, the model performance can be significantly improved over the previous state-of-the-art private baselines. Moreover, we study different ways to obtain priors, which when used with RRWithPrior can additionally improve the model performance, further reducing the accuracy gap between private and non-private models. We complement the empirical results with theoretical analysis showing that LabelDP is provably easier than protecting both the inputs and labels.

Most works in learning with differential privacy (DP) have focused on the setting where each user has a single sample. In this work, we consider the setting where each user holds $m$ samples and the privacy protection is enforced at the level of each user's data. We show that, in this setting, we may learn with a much fewer number of users. Specifically, we show that, as long as each user receives sufficiently many samples, we can learn any privately learnable class via an $(\epsilon, \delta)$-DP algorithm using only $O(\log(1/\delta)/\epsilon)$ users. For $\epsilon$-DP algorithms, we show that we can learn using only $O_{\epsilon}(d)$ users even in the local model, where $d$ is the probabilistic representation dimension. In both cases, we show a nearly-matching lower bound on the number of users required.A crucial component of our results is a generalization of global stability [Bun, Livni, Moran, FOCS 2020] that allows the use of public randomness. Under this relaxed notion, we employ a correlated sampling strategy to show that the global stability can be boosted to be arbitrarily close to one, at a polynomial expense in the number of samples.

We consider the following variant of contextual linear bandits motivated by routing applications in navigational engines and recommendation systems. We wish to learn a hidden $d$-dimensional value $w^*$. Every round, we are presented with a subset $\mathcal{X}_t \subseteq \mathbb{R}^d$ of possible actions. If we choose (i.e. recommend to the user) action $x_t$, we obtain utility $\langle x_t, w^* \rangle$ but only learn the identity of the best action $\arg\max_{x \in \X_t} \langle x, w^* \rangle$.We design algorithms for this problem which achieve regret $O(d\log T)$ and $\exp(O(d \log d))$. To accomplish this, we design novel cutting-plane algorithms with low “regret” -- the total distance between the true point $w^*$ and the hyperplanes the separation oracle returns. We also consider the variant where we are allowed to provide a list of several recommendations. In this variant, we give an algorithm with $O(d^2 \log d)$ regret and list size $\poly(d)$. Finally, we construct nearly tight algorithms for a weaker variant of this problem where the learner only learns the identity of an action that is better than the recommendation. Our results rely on new algorithmic techniques in convex geometry (including a variant of Steiner’s formula for the centroid of a convex set) which may be of independent interest.

In this paper we prove that the sample complexity of properly learning a class of Littlestone dimension d with approximate differential privacy is Õ(d6), ignoring privacy and accuracy parameters. This result answers a question of Bun et al. (FOCS 2020) by improving upon their upper bound of 2O(d) on the sample complexity. Prior to our work, finiteness of the sample complexity for privately learning a class of finite Littlestone dimension was only known for improper private learners, and the fact that our learner is proper answers another question of Bun et al., which was also asked by Bousquet et al. (NeurIPS 2020). Using machinery developed by Bousquet et al., we then show that the sample complexity of sanitizing a binary hypothesis class is at most polynomial in its Littlestone dimension and dual Littlestone dimension. This implies that a class is sanitizable if and only if it has finite Littlestone dimension. An important ingredient of our proofs is a new property of binary hypothesis classes that we call irreducibility, which may be of independent interest.

The -Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over , which can be stated as follows: given a generator matrix and an integer , determine whether the code generated by has distance at most , or, in other words, whether there is a nonzero vector such that has at most nonzero coordinates. The question of whether -Even Set is fixed parameter tractable (FPT) parameterized by the distance has been repeatedly raised in the literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows [1999]. In this work, we show that -Even Set is W [1]-hard under randomized reductions. We also consider the parameterized -Shortest Vector Problem (SVP) , in which we are given a lattice whose basis vectors are integral and an integer , and the goal is to determine whether the norm of the shortest vector (in the norm for some fixed ) is at most . Similar to -Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any , -SVP is W [1]-hard to approximate (under randomized reductions) to some constant factor.

We study a resource allocation setting where $m$ discrete items are to be divided among $n$ agents with additive utilities, and the agents' utilities for individual items are drawn at random from a probability distribution. Since common fairness notions like envy-freeness and proportionality cannot always be satisfied in this setting, an important question is when allocations satisfying these notions exist. In this paper, we close several gaps in the line of work on asymptotic fair division. First, we prove that the classical round-robin algorithm is likely to produce an envy-free allocation provided that $m=\Omega(n\log n/\log\log n)$, matching the lower bound from prior work. We then show that a proportional allocation exists with high probability as long as $m\geq n$, while an allocation satisfying envy-freeness up to any item (EFX) is likely to be present for any relation between $m$ and $n$. Finally, we consider a related setting where each agent is assigned exactly one item and the remaining items are left unassigned, and show that the transition from nonexistence to existence with respect to envy-free assignments occurs at $m=en$.

The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ D. It is known that this problem is notoriously hard, as there is no k 1/4− o (1) -approximation algorithm under Gap-ETH, even when parametrizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k . For the bi -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution in a bidirected graph G , i.e., for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k . We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k .

Differential privacy (DP) is a formal notion for quantifying the privacy loss of algorithms. Algorithms in the central model of DP achieve high accuracy but make the strongest trust assumptions whereas those in the local DP model make the weakest trust assumptions but incur substantial accuracy loss. The shuffled DP model [Bittau et al 2017, Erlingsson et al 2019, Cheu et al 19] has recently emerged as a feasible middle ground between the central and local models, providing stronger trust assumptions than the former while promising higher accuracies than the latter. In this paper, we obtain practical communication-efficient algorithms in the shuffled DP model for two basic aggregation primitives used in machine learning: 1) binary summation, and 2) histograms over a moderate number of buckets. Our algorithms achieve accuracy that is arbitrarily close to that of central DP algorithms with an expected communication per user essentially matching what is needed without any privacy constraints! We demonstrate the practicality of our algorithms by experimentally evaluating them and comparing their performance to several widely-used protocols such as Randomized Response [Warner 1965] and RAPPOR [Erlingsson et al. 2014].

In approval-based multiwinner elections, we are given a set of voters, a set of candidates, and, for each voter, a set of candidates approved by the voter. The goal is to find a committee of size k that maximizes the total utility of the voters. In this paper, we study approximability of Thiele rules, which are known to be NP-hard to solve exactly. We provide a tight polynomial time approximation algorithm for a natural class of geometrically dominant weights that includes such voting rules as Proportional Approval Voting or p-Geometric. The algorithm is relatively simple: first we solve a linear program and then we round a solution by employing a framework called pipage rounding due to Ageev and Sviridenko (2004) and Calinescu et al. (2011). We provide a matching lower bound via a reduction from the Label Cover problem. Moreover, assuming a conjecture called Gap-ETH, we show that better approximation ratio cannot be obtained even in time f(k)*pow(n,o(k)).

We study the computational complexity of adversarially robust proper learning of halfspaces in the distribution-independent agnostic PAC model, with a focus on $L_p$ perturbations. We give a computationally efficient learning algorithm and a nearly matching computational hardness result for this problem. An interesting implication of our findings is that the $L_{\infty}$ perturbations case is provably computationally harder than the case $2 \leq p < \infty$.

We study the task of differentially private clustering. For several basic clustering problems, including Euclidean DensestBall, 1-Cluster, k-means, and k-median, we give efficient differentially private algorithms that achieve essentially the same approximation ratios as those that can be obtained by any non-private algorithm, while incurring only small additive errors. This improves upon existing efficient algorithms that only achieve some large constant approximation factors. Our results also imply an improved algorithm for the Sample and Aggregate privacy framework. Furthermore, we show that one of the tools used in our 1-Cluster algorithm can be employed to get a faster quantum algorithm for ClosestPair in a moderate number of dimensions.

We show, assuming the (randomized) Gap Exponential Time Hypothesis (Gap-ETH), that the following tasks cannot be done in T(k) · No(k)-time for any function T where N denote the input size: -approximation for Max k-Coverage for any constant ɛ > 0, -approximation for k-Median (in general metrics) for any constant ɛ > 0. -approximation for k-Mean (in general metrics) for any constant ɛ > 0. Any constant factor approximation for k-Unique Set Cover, k-Nearest Codeword Problem and k-Closest Vector Problem. (1 + δ)-approximation for k-Minimum Distance Problem and k-Shortest Vector Problem for some δ > 0. Since all problems considered here can be trivially solved in NO(k) time, our running time lower bounds are tight up to a constant factor in the exponent. In terms of approximation ratios, Max k-Coverage is well-known to admit polynomial-time -approximation algorithms, and, recently, it was shown that k-Median and k-Median are approximable to within factors of and respectively in FPT time [20]; hence, our inapproximability ratios are also tight for these three problems. For the remaining problems, no non-trivial FPT approximation algorithms are known. The starting point of all our hardness results is the Label Cover problem (with projection constraints). We show that Label Cover cannot be approximated to within any constant factor in T(k) · No(k) time, where N and k denote the size of the input and the number of nodes on the side with the larger alphabet respectively. With this hardness, the above results follow immediately from known reductions. The hardness of Label Cover is in turn shown via a t-wise agreement testing theorem of the following form: given local boolean functions f1, …,fk on domains S1, …, Sk ⊆ [n], if random t functions “weakly agree” with sufficiently large probability, then we can find a global boolean function g: [n] → {0, 1} that “mostly agrees” with “many” of the local functions. We prove such a statement in the regime where S1, …, Sk are “random-looking” sets of size Θ(n/k).

Consensus halving refers to the problem of dividing a resource into two parts so that every agent values both parts equally. Prior work shows that, when the resource is represented by an interval, a consensus halving with at most n cuts always exists but is hard to compute even for agents with simple valuation functions. In this paper, we study consensus halving in a natural setting in which the resource consists of a set of items without a linear ordering. For agents with linear and additively separable utilities, we present a polynomial-time algorithm that computes a consensus halving with at most n cuts and show that n cuts are almost surely necessary when the agents’ utilities are randomly generated. On the other hand, we show that, for a simple class of monotonic utilities, the problem already becomes polynomial parity argument, directed version–hard. Furthermore, we compare and contrast consensus halving with the more general problem of consensus k-splitting, with which we wish to divide the resource into k parts in possibly unequal ratios and provide some consequences of our results on the problem of computing small agreeable sets.

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable (FPT) algorithms. The questions, which have been asked several times, are whether there is a nontrivial FPT-approximation algorithm for the Maximum Clique $({\sf Clique})$ and Minimum Dominating Set $({\sf DomSet})$ problems parameterized by the size of the optimal solution. In particular, letting ${\sf OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t({\sf OPT}){\operatorname{poly}}(N)$ time and outputs a solution of size $f({\sf OPT})$ for any computable functions $t$ and $f$ that are independent of $N$ (for ${\sf Clique}$, we want $f({\sf OPT})=\omega(1)$)? In this paper, we show that both ${\sf Clique}$ and ${\sf DomSet}$ admit no nontrivial FPT-approximation algorithm, i.e., there is no $o({\sf OPT})$-FPT-approximation algorithm for ${\sf Clique}$ and no $f({\sf OPT})$-FPT-approximation algorithm for ${\sf DomSet}$ for any function $f$. In fact, our results imply something even stronger: The best way to solve ${\sf Clique}$ and ${\sf DomSet}$, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis [I. Dinur. ECCC, TR16-128, 2016; P. Manurangsi and P. Raghavendra, preprint, arXiv:1607.02986, 2016], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3 \sf SAT formula and one which is not even $(1 - \varepsilon)$-satisfiable for some constant $\varepsilon > 0$. Besides ${\sf Clique}$ and ${\sf DomSet}$, we also rule out nontrivial FPT-approximation for the Maximum Biclique problem, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs, and we rule out the $k^{o(1)}$-FPT-approximation algorithm for the Densest $k$-Subgraph problem.

We consider the problem of determining the maximal $\alpha \in (0,1]$ such that every matching $M$ of size $k$ (or at most $k$) in a bipartite graph $G$ contains an induced matching of size at least $\alpha |M|$. This measure was recently introduced in [N. Alon et al., Adv. Neural Inf. Process. Syst., 2017, pp. 2097--2106] and is motivated by computational models in cognitive neuroscience as well as by modeling interference in radio and communication networks. We prove various hardness results for computing $\alpha$ either exactly or approximately. En route to our results, we also consider the maximum connected matching problem: determining the largest matching $N$ in a graph $G$ such that every two edges in $N$ are connected by an edge. We prove a nearly optimal $n^{1-\epsilon}$ hardness of approximation result (under randomized reductions) for connected matching in bipartite graphs (with both sides of cardinality $n$). Toward this end we define bipartite half-covers: a new combinatorial object that may be of independent interest. To our knowledge, the best previous hardness result for the maximum connected matching problem was that it is hard to approximate within some constant $\beta>1$. Finally, we demonstrate the existence of bipartite graphs with $n$ vertices on each side of average degree $d$, achieving $\alpha=1/2-\epsilon$ for matchings of size sufficiently smaller than $n/d$. This nearly matches the trivial upper bound of $1/2$ on $\alpha$ which holds for any graph containing a path of length 3.

We consider a fair division setting in which $m$ indivisible items are to be allocated among $n$ agents, where the agents have additive utilities and the agents' utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when $m=\Omega(n\log n)$ but not when $m=n+o(n)$, and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition---instead, the transition is governed by the divisibility relation between $m$ and $n$. On the one hand, if $m$ is divisible by $n$, an envy-free allocation exists with high probability as long as $m\geq 2n$. On the other hand, if $m$ is not “almost” divisible by $n$, an envy-free allocation is unlikely to exist even when $m=\Theta(n\log n/\log\log n)$.

In the SHIFT-BRIBERY problem we are given an election, a preferred candidate, and the costs of shifting this preferred candidate up the voters’ preference orders. The goal is to find such a set of shifts that ensures that the preferred candidate wins the election. We give the first polynomial-time approximation scheme for the case of positional scoring rules, and for the Copeland rule we show strong inapproximability results.

We consider a fair division setting in which m indivisible items are to be allocated among n agents, where the agents have additive utilities and the agents’ utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when m = Ω (n log n) but not when m = n + o (n), and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition— instead, the transition is governed by the divisibility relation between m and n. On the one hand, if m is divisible by n, an envy-free allocation exists with high probability as long as m ≥ 2n. On the other hand, if m is not “almost” divisible by , an envy-free allocation is unlikely to exist even when m = Θ(n log n)/log log n).

We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions.

Given a set of n points in R^d, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the l_p-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when d=omega(log n) was raised as an open question in recent works (Abboud-Rubinstein-Williams [FOCS'17], Williams [SODA'18], David-Karthik-Laekhanukit [SoCG'18]). In this paper, we show that for every p in R_{>= 1} cup {0}, under the Strong Exponential Time Hypothesis (SETH), for every epsilon>0, the following holds: - No algorithm running in time O(n^{2-epsilon}) can solve the Closest Pair problem in d=(log n)^{Omega_{epsilon}(1)} dimensions in the l_p-metric. - There exists delta = delta(epsilon)>0 and c = c(epsilon)>= 1 such that no algorithm running in time O(n^{1.5-epsilon}) can approximate Closest Pair problem to a factor of (1+delta) in d >= c log n dimensions in the l_p-metric. In particular, our first result is shown by establishing the computational equivalence of the bichromatic Closest Pair problem and the (monochromatic) Closest Pair problem (up to n^{epsilon} factor in the running time) for d=(log n)^{Omega_epsilon(1)} dimensions. Additionally, under SETH, we rule out nearly-polynomial factor approximation algorithms running in subquadratic time for the (monochromatic) Maximum Inner Product problem where we are given a set of n points in n^{o(1)}-dimensional Euclidean space and are required to find a pair of distinct points in the set that maximize the inner product. At the heart of all our proofs is the construction of a dense bipartite graph with low contact dimension, i.e., we construct a balanced bipartite graph on n vertices with n^{2-epsilon} edges whose vertices can be realized as points in a (log n)^{Omega_epsilon(1)}-dimensional Euclidean space such that every pair of vertices which have an edge in the graph are at distance exactly 1 and every other pair of vertices are at distance greater than 1. This graph construction is inspired by the construction of locally dense codes introduced by Dumer-Miccancio-Sudan [IEEE Trans. Inf. Theory'03].

We study the problem of {\em properly} learning large margin halfspaces in the agnostic PAC model. In more detail, we study the complexity of properly learning $d$-dimensional halfspaces on the unit ball within misclassification error $\alpha \cdot \opt_{\gamma} + \eps$, where $\opt_{\gamma}$ is the optimal $\gamma$-margin error rate and $\alpha \geq 1$ is the approximation ratio. We give learning algorithms and computational hardness results for this problem, for all values of the approximation ratio $\alpha \geq 1$, that are nearly-matching for a range of parameters. Specifically, for the natural setting that $\alpha$ is any constant bigger than one, we provide an essentially tight complexity characterization. On the positive side, we give an $\alpha = 1.01$-approximate proper learner that uses $O(1/(\eps^2\gamma^2))$ samples (which is optimal) and runs in time $\poly(d/\eps) \cdot 2^{\tilde{O}(1/\gamma^2)}$. On the negative side, we show that {\em any} constant factor approximate proper learner has runtime $\poly(d/\eps) \cdot 2^{(1/\gamma)^{2-o(1)}}$, assuming the Exponential Time Hypothesis.

We study the problem of deleting the smallest set S of vertices (resp. edges) from a given graph G such that the induced subgraph (resp. subgraph) G\S belongs to some class ℋ. We consider the case where graphs in ℋ have treewidth bounded by t, and give a general framework to obtain approximation algorithms for both vertex and edge-deletion settings from approximation algorithms for certain natural graph partitioning problems called k-Subset Vertex Separator and k-Subset Edge Separator, respectively. For the vertex deletion setting, our framework combined with the current best result for k-Subset Vertex Separator, improves approximation ratios for basic problems such as k-Treewidth Vertex Deletion and Planar-ℱ Vertex Deletion. Our algorithms are simpler than previous works and give the first deterministic and uniform approximation algorithms under the natural parameterization. For the edge deletion setting, we give improved approximation algorithms for k-Subset Edge Separator combining ideas from LP relaxations and important separators. We present their applications in bounded-degree graphs, and also give an APX-hardness result for the edge deletion problems.

We study the parameterized complexity of approximating the k -Dominating Set (DomSet) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F ( k ) ⋅ k whenever the graph G has a dominating set of size k . When such an algorithm runs in time T ( k ) ⋅ poly ( n ) (i.e., FPT-time) for some computable function T , it is said to be an F ( k )- FPT-approximation algorithm for k -DomSet. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the “most infamous” open problems in parameterized complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1] ≠ FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T , F and every constant ε > 0: • Assuming W[1] ≠ FPT, there is no F ( k )- FPT-approximation algorithm for k -DomSet. • Assuming the Exponential Time Hypothesis (ETH), there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n o ( k ) time. • Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n k − ε time. • Assuming the k -SUM Hypothesis, for every integer k ≥ 3, there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n ⌈ k /2 ⌉ − ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under sf W[1] ≠ FPT and (log 1/4 &minus ε k )-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F ( k )-FPT-approximation algorithm for any function F was shown under Gap-ETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form n &delta k for any absolute constant δ > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well-studied problem or a variant of one; this allows us to easily apply known techniques to solve them.

The log-density method is a powerful algorithmic framework which in recent years has given rise to the best-known approximations for a variety of problems, including Densest-k-Subgraph and Small Set Bipartite Vertex Expansion. These approximations have been conjectured to be optimal based on various instantiations of a general conjecture: that it is hard to distinguish a fully random combinatorial structure from one which contains a similar planted sub-structure with the same "log-density". We bolster this conjecture by showing that in a random hypergraph with edge probability n^{-alpha}, Omega(log n) rounds of Sherali-Adams cannot rule out the existence of a k-subhypergraph with edge density k^{-alpha-o(1)}, for any k and alpha. This holds even when the bound on the objective function is lifted. This gives strong integrality gaps which exactly match the gap in the above distinguishing problems, as well as the best-known approximations, for Densest k-Subgraph, Smallest p-Edge Subgraph, their hypergraph extensions, and Small Set Bipartite Vertex Expansion (or equivalently, Minimum p-Union). Previously, such integrality gaps were known only for Densest k-Subgraph for one specific parameter setting.

In this work, we study the trade-off between the running time of approximation algorithms and their approximation guarantees. By leveraging a structure of the `hard' instances of the Arora-Rao-Vazirani lemma [JACM'09], we show that the Sum-of-Squares hierarchy can be adapted to provide `fast', but still exponential time, approximation algorithms for several problems in the regime where they are believed to be NP-hard. Specifically, our framework yields the following algorithms; here $n$ denote the number of vertices of the graph and $r$ can be any positive real number greater than 1 (possibly depending on $n$). (i) A $\left(2 - \frac{1}{O(r)}\right)$-approximation algorithm for Vertex Cover that runs in $\exp\left(\frac{n}{2^{r^2}}\right)n^{O(1)}$ time. (ii) An $O(r)$-approximation algorithms for Uniform Sparsest Cut, Balanced Separator, Minimum UnCut and Minimum 2CNF Deletion that runs in $\exp\left(\frac{n}{2^{r^2}}\right)n^{O(1)}$ time. Our algorithm for Vertex Cover improves upon Bansal et al.'s algorithm [arXiv:1708.03515] which achieves $\left(2 - \frac{1}{O(r)}\right)$-approximation in time $\exp\left(\frac{n}{r^r}\right)n^{O(1)}$. For the remaining problems, our algorithms improve upon $O(r)$-approximation $\exp\left(\frac{n}{2^r}\right)n^{O(1)}$-time algorithms that follow from a work of Charikar et al. [SIAM J. Comput.'10].

Consider the following asynchronous, opportunistic communication model over a graph $G$: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say $\pm 1$ with probability $1/2$; then, in each round, the two endpoints of the currently active edge update their values to their average. We show that, if $G$ exhibits a two-community structure (for example, two expanders connected by a sparse cut), the values held by the nodes will collectively reflect the underlying community structure over a suitable phase of the above process, allowing efficient and effective recovery in important cases. In more detail, we first provide a first-moment analysis showing that, for a large class of almost-regular clustered graphs that includes the stochastic block model, the expected values held by all but a negligible fraction of the nodes eventually reflect the underlying cut signal. We prove this property emerges after a mixing period of length $\mathcal O(n\log n)$. We further provide a second-moment analysis for a more restricted class of regular clustered graphs that includes the regular stochastic block model. For this case, we are able to show that most nodes can efficiently and locally identify their community of reference over a suitable time window. This results in the first opportunistic protocols that approximately recover community structure using only polylogarithmic work per node. Even for the above class of regular graphs, our second moment analysis requires new concentration bounds on the product of certain random matrices that are technically challenging and possibly of independent interest.

The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E) and a set {D}subseteq V x V of k demand pairs. The aim is to compute the cheapest network N subseteq G for which there is an s - t path for each (s,t)in {D}. It is known that this problem is notoriously hard as there is no k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parameterizing the runtime by k [Dinur Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k. For the bi-DSN_Planar problem, the aim is to compute a planar optimum solution N subseteq G in a bidirected graph G, i.e. for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSN_Planar, unless FPT=W[1]. Additionally we study several generalizations of bi-DSN_Planar and obtain upper and lower bounds on obtainable runtimes parameterized by k. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network N subseteq G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists when parameterized by k [Chitnis et al., IPEC 2013]. We show a tight inapproximability result: under Gap-ETH there is no (2-{epsilon})-approximation algorithm parameterized by k (for any epsilon0). To the best of our knowledge, this is the first example of a W[1]-hard problem admitting a non-trivial parameterized approximation factor which is also known to be tight! Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k.

The k-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over F_2, which can be stated as follows: given a generator matrix A and an integer k, determine whether the code generated by A has distance at most k. Here, k is the parameter of the problem. The question of whether k-Even Set is fixed parameter tractable (FPT) has been repeatedly raised in literature and has earned its place in Downey and Fellows' book (2013) as one of the "most infamous" open problems in the field of Parameterized Complexity. In this work, we show that k-Even Set does not admit FPT algorithms under the (randomized) Gap Exponential Time Hypothesis (Gap-ETH) [Dinur'16, Manurangsi-Raghavendra'16]. In fact, our result rules out not only exact FPT algorithms, but also any constant factor FPT approximation algorithms for the problem. Furthermore, our result holds even under the following weaker assumption, which is also known as the Parameterized Inapproximability Hypothesis (PIH) [Lokshtanov et al.'17]: no (randomized) FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only 0.99-satisfiable (where the parameter is the number of variables). We also consider the parameterized k-Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer k, and the goal is to determine whether the norm of the shortest vector (in the l_p norm for some fixed p) is at most k. Similar to k-Even Set, this problem is also a long-standing open problem in the field of Parameterized Complexity. We show that, for any p > 1, k-SVP is hard to approximate (in FPT time) to some constant factor, assuming PIH. Furthermore, for the case of p = 2, the inapproximability factor can be amplified to any constant.

We study 2-ary constraint satisfaction problems (2-CSPs), which can be stated as follows: given a constraint graph G = (V,E), an alphabet set Sigma and, for each edge {u, v}, a constraint C_uv, the goal is to find an assignment sigma from V to Sigma that satisfies as many constraints as possible, where a constraint C_uv is said to be satisfied by sigma if C_uv contains (sigma(u),sigma(v)). While the approximability of 2-CSPs is quite well understood when the alphabet size |Sigma| is constant (see e.g. [37]), many problems are still open when |Sigma| becomes super constant. One open problem that has received significant attention in the literature is whether it is hard to approximate 2-CSPs to within a polynomial factor of both |Sigma| and |V| (i.e. (|Sigma||V|)^Omega(1) factor). As a special case of the so-called Sliding Scale Conjecture, Bellare et al. [5] suggested that the answer to this question might be positive. Alas, despite many efforts by researchers to resolve this conjecture (e.g. [39, 4, 20, 21, 35]), it still remains open to this day. In this work, we separate |V| and |Sigma| and ask a closely related but weaker question: is it hard to approximate 2-CSPs to within a polynomial factor of |V| (while |Sigma| may be super-polynomial in |Sigma|)? Assuming the exponential time hypothesis (ETH), we answer this question positively: unless ETH fails, no polynomial time algorithm can approximate 2-CSPs to within a factor of |V|^{1-o(1)}. Note that our ratio is not only polynomial but also almost linear. This is almost optimal since a trivial algorithm yields an O(|V|)-approximation for 2-CSPs. Thanks to a known reduction [25, 16] from 2-CSPs to the Directed Steiner Network (DSN) problem, our result implies an inapproximability result for the latter with polynomial ratio in terms of the number of demand pairs. Specifically, assuming ETH, no polynomial time algorithm can approximate DSN to within a factor of k^{1/4 - o(1)} where k is the number of demand pairs. The ratio is roughly the square root of the approximation ratios achieved by best known polynomial time algorithms [15, 26], which yield O(k^{1/2 + epsilon})-approximation for every constant epsilon > 0. Additionally, under Gap-ETH, our reduction for 2-CSPs not only rules out polynomial time algorithms, but also fixed parameter tractable (FPT) algorithms parameterized by the number of variables |V|. These are algorithms with running time g(|V|)·|Sigma|^O(1) for some function g. Similar improvements apply for DSN parameterized by the number of demand pairs k.

We study the parameterized complexity of approximating the $k$-Dominating Set (DomSet) problem where an integer $k$ and a graph $G$ on $n$ vertices are given as input, and the goal is to find a dominating set of size at most $F(k) \cdot k$ whenever the graph $G$ has a dominating set of size $k$. When such an algorithm runs in time $T(k) \cdot poly(n)$ (i.e., FPT-time) for some computable function $T$, it is said to be an $F(k)$-FPT-approximation algorithm for $k$-DomSet. We prove the following for every computable functions $T, F$ and every constant $\varepsilon > 0$: $\bullet$ Assuming $W[1]\neq FPT$, there is no $F(k)$-FPT-approximation algorithm for $k$-DomSet. $\bullet$ Assuming the Exponential Time Hypothesis (ETH), there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{o(k)}$ time. $\bullet$ Assuming the Strong Exponential Time Hypothesis (SETH), for every integer $k \geq 2$, there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{k - \varepsilon}$ time. $\bullet$ Assuming the $k$-Sum Hypothesis, for every integer $k \geq 3$, there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{\lceil k/2 \rceil - \varepsilon}$ time. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well studied problem or a variant of one; this allows us to easily apply known techniques to solve them.

We study the complexity of computing the VC Dimension and Littlestone’s Dimension. Given an explicit description of a finite universe and a concept class (a binary matrix whose $(x,C)$-th entry is $1$ iff element $x$ belongs to concept $C$), both can be computed exactly in quasi-polynomial time ($n^O(\log n)$). Assuming the randomized Exponential Time Hypothesis (ETH), we prove nearly matching lower bounds on the running time, that hold even for \em approximation algorithms.

We consider questions that arise from the intersection between the areas of approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [1], [2], [3]) are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting OPT be the optimum and N be the size of the input, is there an algorithm that runs in t(OPT) poly(N) time and outputs a solution of size f(OPT), for any functions t and f that are independent of N (for Clique, we want f(OPT) = ω(1))? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no o(OPT)-FPT-approximation algorithm for Clique and no f(OPT)-FPT-approximation algorithm for DomSet, for any function f (e.g., this holds even if f is an exponential or the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (GapETH) [4], [5], which states that no 2o(n)-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1 - ε)-satisfiable for some constant ε > 0. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs. Previously only exact versions of these problems were known to be W[1]-hard [6], [7], [8]. Additionally, we rule out ko(1)-FPT-approximation algorithm for Densest k-Subgraph although this ratio does not yet match the trivial O(k)-approximation algorithm. To the best of our knowledge, prior results only rule out constant factor approximation for Clique [9], [10] and log1/4+ε(OPT) approximation for DomSet for any constant ε > 0 [11]. Our result on Clique significantly improves on [9], [10]. However, our result on DomSet is incomparable to [11] since their results hold under ETH while our results hold under Gap-ETH, which is a stronger assumption.

The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small set of vertices whose expansion is almost zero and one in which all small sets of vertices have expansion almost one. In this work, we prove conditional inapproximability results for the following graph problems based on this hypothesis: - Maximum Edge Biclique (MEB): given a bipartite graph G, find a complete bipartite subgraph of G with maximum number of edges. We show that, assuming SSEH and that NP != BPP, no polynomial time algorithm gives n^{1 - epsilon}-approximation for MEB for every constant epsilon > 0. - Maximum Balanced Biclique (MBB): given a bipartite graph G, find a balanced complete bipartite subgraph of G with maximum number of vertices. Similar to MEB, we prove n^{1 - epsilon} ratio inapproximability for MBB for every epsilon > 0, assuming SSEH and that NP != BPP. - Minimum k-Cut: given a weighted graph G, find a set of edges with minimum total weight whose removal splits the graph into k components. We prove that this problem is NP-hard to approximate to within (2 - epsilon) factor of the optimum for every epsilon > 0, assuming SSEH. The ratios in our results are essentially tight since trivial algorithms give n-approximation to both MEB and MBB and 2-approximation algorithms are known for Minimum k-Cut [Saran and Vazirani, SIAM J. Comput., 1995]. Our first two results are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani [Raghavendra et al., CCC, 2012] to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test [Bansal and Khot, FOCS, 2009] whereas our last result is shown via an elementary reduction.

A $(k \times l)$-birthday repetition $\mathcal{G}^{k \times l}$ of a two-prover game $\mathcal{G}$ is a game in which the two provers are sent random sets of questions from $\mathcal{G}$ of sizes $k$ and $l$ respectively. These two sets are sampled independently uniformly among all sets of questions of those particular sizes. We prove the following birthday repetition theorem: when $\mathcal{G}$ satisfies some mild conditions, $val(\mathcal{G}^{k \times l})$ decreases exponentially in $Ω(kl/n)$ where $n$ is the total number of questions. Our result positively resolves an open question posted by Aaronson, Impagliazzo and Moshkovitz (CCC 2014). As an application of our birthday repetition theorem, we obtain new fine-grained hardness of approximation results for dense CSPs. Specifically, we establish a tight trade-off between running time and approximation ratio for dense CSPs by showing conditional lower bounds, integrality gaps and approximation algorithms. In particular, for any sufficiently large $i$ and for every $k \geq 2$, we show the following results: - We exhibit an $O(q^{1/i})$-approximation algorithm for dense Max $k$-CSPs with alphabet size $q$ via $O_k(i)$-level of Sherali-Adams relaxation. - Through our birthday repetition theorem, we obtain an integrality gap of $q^{1/i}$ for $\tildeΩ_k(i)$-level Lasserre relaxation for fully-dense Max $k$-CSP. - Assuming that there is a constant $ε> 0$ such that Max 3SAT cannot be approximated to within $(1-ε)$ of the optimal in sub-exponential time, our birthday repetition theorem implies that any algorithm that approximates fully-dense Max $k$-CSP to within a $q^{1/i}$ factor takes $(nq)^{\tilde Ω_k(i)}$ time, almost tightly matching the algorithmic result based on Sherali-Adams relaxation.

We study the problem of finding a small subset of items that is agreeable to all agents, meaning that all agents value the subset at least as much as its complement. Previous work has shown worst-case bounds, over all instances with a given number of agents and items, on the number of items that may need to be included in such a subset. Our goal in this paper is to efficiently compute an agreeable subset whose size approximates the size of the smallest agreeable subset for a given instance. We consider three well-known models for representing the preferences of the agents: ordinal preferences on single items, the value oracle model, and additive utilities. In each of these models, we establish virtually tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time.

Many known optimal NP-hardness of approximation results are reductions from a problem called Label Cover. The input is a bipartite graph G = (L, R, E) and each edge e = (x,y) ∊ E carries a projection π∊ that maps labels to x to labels to y. The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio efficiently achievable for Label-Cover is of the form N−c where N = nk, n is the number of vertices, k is the number of labels, and 0 ≤ c < 1 is some constant. Inspired by a framework originally developed for Densest k-SuBGRAPH, we propose a “log density threshold” for the approximability of Label-Cover. Specifically, we suggest the possibility that the Label-Cover approximation problem undergoes a computational phase transition at the same threshold at which local algorithms for its random counterpart fail. This threshold is We then design, for any ∊ > 0, a polynomial-time approximation algorithm for semirandom Label-Cover whose approximation ratio is In our semi-random model, the input graph is random (or even just expanding), and the projections on the edges are arbitrary. For worst-case Label-Cover we show a polynomial- time algorithm whose approximation ratio is roughly Ν-°·233. The previous best efficient approximation ratio was Ν−0·25. We present some evidence towards an Ν−c threshold by constructing integrality gaps for Νω(1) rounds of the Sum-of-squares/Lasserre hierarchy of the natural relaxation of Label Cover. For general 2CSP the “log density threshold” is Ν−0·25, and we give a polynomial-time algorithm in the semi-random model whose approximation ratio is Ν−0·25+∊ for any ∊ > 0.

In the Densest k-Subgraph (DkS) problem, given an undirected graph G and an integer k, the goal is to find a subgraph of G on k vertices that contains maximum number of edges. Even though Bhaskara et al.'s state-of-the-art algorithm for the problem achieves only O(n1/4 + ϵ) approximation ratio, previous attempts at proving hardness of approximation, including those under average case assumptions, fail to achieve a polynomial ratio; the best ratios ruled out under any worst case assumption and any average case assumption are only any constant (Raghavendra and Steurer) and 2O(log2/3 n) (Alon et al.) respectively.

In this paper, we prove an almost-optimal hardness for Max k-CSP_R based on Khot's Unique Games Conjecture (UGC). In Max k-CSP_R, we are given a set of predicates each of which depends on exactly k variables. Each variable can take any value from 1, 2, ..., R. The goal is to find an assignment to variables that maximizes the number of satisfied predicates. Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max k-CSP_R to within factor 2^{O(k log k)}(log R)^{k/2}/R^{k - 1} for any k, R. To the best of our knowledge, this result improves on all the known hardness of approximation results when 3 <= k = o(log R/log log R). In this case, the previous best hardness result was NP-hardness of approximating within a factor O(k/R^{k-2}) by Chan. When k = 2, our result matches the best known UGC-hardness result of Khot, Kindler, Mossel and O'Donnell. In addition, by extending an algorithm for Max 2-CSP_R by Kindler, Kolla and Trevisan, we provide an Omega(log R/R^{k - 1})-approximation algorithm for Max k-CSP_R. This algorithm implies that our inapproximability result is tight up to a factor of 2^{O(k \log k)}(\log R)^{k/2 - 1}. In comparison, when 3 <= k is a constant, the previously known gap was $O(R)$, which is significantly larger than our gap of O(polylog R). Finally, we show that we can replace the Unique Games Conjecture assumption with Khot's d-to-1 Conjecture and still get asymptotically the same hardness of approximation.

In this paper, we present a polynomial-time algorithm that approximates sufficiently high-value Max 2-CSPs on sufficiently dense graphs to within O(N^epsilon) approximation ratio for any constant epsilon > 0. Using this algorithm, we also achieve similar results for free games, projection games on sufficiently dense random graphs, and the Densest k-Subgraph problem with sufficiently dense optimal solution. Note, however, that algorithms with similar guarantees to the last algorithm were in fact discovered prior to our work by Feige et al. and Suzuki and Tokuyama. In addition, our idea for the above algorithms yields the following by-product: a quasi-polynomial time approximation scheme (QPTAS) for satisfiable dense Max 2-CSPs with better running time than the known algorithms.