In this work we consider the Metric Steiner Forest problem in the sublinear time model. Given a set \(V\) of \(n\) points in a metric space where distances are provided by means of query access to an \(n \times n\) distance matrix, along with a set of \(k\) terminal pairs (\((s_1, t_1), \ldots, (s_k, t_k), \in V \times V\), the goal is to find a minimum-weight subset of edges that connects each terminal pair. Although sublinear time algorithms have been studied for estimating the weight of a minimum spanning tree in both general and metric settings, as well as for the metric Steiner Tree problem, no sublinear time algorithm was known for the metric Steiner Forest problem.

The goal in the stochastic vertex cover problem is to obtain an approximately minimum vertex cover for a graph G⋆ that is realized by sampling each edge independently with some probability p∈ (0, 1] in a base graph G = (V, E). The algorithm is given the base graph G and the probability p as inputs, but its only access to the realized graph G⋆ is through queries on individual edges in G that reveal the existence (or not) of the queried edge in G⋆. In this paper, we resolve the central open question for this problem: to find a (1+ε)-approximate vertex cover using only Oε(n/p) edge queries. Prior to our work, there were two incomparable state-of-the-art results for this problem: a (3/2+ε)-approximation using Oε(n/p) queries (Derakhshan, Durvasula, and Haghtalab, 2023) and a (1+ε)-approximation using Oε((n/p)· RS(n)) queries (Derakhshan, Saneian, and Xun, 2025), where RS(n) is known to be at least 2Ω(logn/loglogn) and could be as large as n/2Θ(log* n). Our improved upper bound of Oε(n/p) matches the known lower bound of Ω(n/p) for any constant-factor approximation algorithm for this problem (Behnezhad, Blum, and Derakhshan, 2022). A key tool in our result is a new concentration bound for the size of minimum vertex cover on random graphs, which might be of independent interest.

We consider the Survivable Network Design problem (SNDP) in the single-pass insertion-only streaming model. The input to SNDP is an edge-weighted graph G = (V, E) and an integer connectivity requirement r(uv) for each u, v ∈ V. The objective is to find a minimum-weight subgraph H ⊆ G such that, for every pair of vertices u, v ∈ V, u and v are r(uv)-edge/vertex-connected. Recent work by [Ce Jin et al., 2024] obtained approximation algorithms for edge-connectivity augmentation, and via that, also derived algorithms for edge-connectivity SNDP (EC-SNDP). In this work we consider vertex-connectivity setting (VC-SNDP) and obtain several results for it as well as improved results for EC-SNDP. - We provide a general framework for solving connectivity problems including SNDP and others in streaming; this is based on a connection to fault-tolerant spanners. For VC-SNDP we provide an O(tk)-approximation in Õ(k^{1-1/t}n^{1 + 1/t}) space, where k is the maximum connectivity requirement, assuming an exact algorithm at the end of the stream. Using a refined LP-based analysis, we provide an O(β t)-approximation where β is the integrality gap of the natural cut-based LP relaxation. These are the first approximation algorithms in the streaming model for VC-SNDP. When applied to the EC-SNDP, our framework provides an O(t)-approximation in Õ(k^{1/2-1/(2t)}n^{1 + 1/t} + kn) space, improving the O(t log k)-approximation of [Ce Jin et al., 2024] using Õ(kn^{1+1/t}) space; this also extends to element-connectivity SNDP. - We consider vertex connectivity-augmentation in the link-arrival model. The input is a k-vertex-connected spanning subgraph G, and additional weighted links L arrive in the stream; the goal is to store the min-weight set of links such that G ∪ L is (k+1)-vertex-connected. We obtain constant-factor approximations in near-linear space for k = 1, 2. Our result for k = 2 is based on using the SPQR tree, a novel application for this well-known representation of 2-connected graphs.

We study the classic Max-Cut problem under multiple cardinality constraints, which we refer to as the Constrained Max-Cut problem. Given a graph G = (V, E), a partition of the vertices into c disjoint parts V₁, …, V_c, and cardinality parameters k₁, …, k_c, the goal is to select a set S ⊆ V such that |S ∩ V_i| = k_i for each i ∈ [c], maximizing the total weight of edges crossing S (i.e., edges with exactly one endpoint in S). By designing an approximate kernel for Constrained Max-Cut and building on the correlation rounding technique of Raghavendra and Tan (2012), we present a (0.858 - ε)-approximation algorithm for the problem when c = O(1). The algorithm runs in time O(min{k/ε, n}^poly(c/ε) + poly(n)), where k = ∑_{i∈[c]} k_i and n = |V|. This improves upon the (1/2 + ε₀)-approximation of Feige and Langberg (2001) for Max-Cut_k (the special case when c = 1, k₁ = k), and generalizes the (0.858 - ε)-approximation of Raghavendra and Tan (2012), which only applies when min{k,n-k} = Ω(n) and does not handle multiple constraints. We also establish that, for general values of c, it is NP-hard to determine whether a feasible solution exists that cuts all edges. Finally, we present a 1/2-approximation algorithm for Max-Cut under an arbitrary matroid constraint.

In this paper we study constrained subspace approximation problem. Given a set of n points {a₁,…,a_n} in ℝ^d, the goal of the subspace approximation problem is to find a k dimensional subspace that best approximates the input points. More precisely, for a given p ≥ 1, we aim to minimize the pth power of the 𝓁_p norm of the error vector (‖a₁-Pa₁‖,…,‖a_n-Pa_n‖), where P denotes the projection matrix onto the subspace and the norms are Euclidean. In constrained subspace approximation (CSA), we additionally have constraints on the projection matrix P. In its most general form, we require P to belong to a given subset 𝒮 that is described explicitly or implicitly. We introduce a general framework for constrained subspace approximation. Our approach, that we term coreset-guess-solve, yields either (1+ε)-multiplicative or ε-additive approximations for a variety of constraints. We show that it provides new algorithms for partition-constrained subspace approximation with applications to fair subspace approximation, k-means clustering, and projected non-negative matrix factorization, among others. Specifically, while we reconstruct the best known bounds for k-means clustering in Euclidean spaces, we improve the known results for the remainder of the problems.

In the Markov paging model, one assumes that page requests are drawn from a Markov chain over the pages in memory, and the goal is to maintain a fast cache that suffers few page faults in expectation. While computing the optimal online algorithm $(\mathrm{OPT})$ for this problem naively takes time exponential in the size of the cache, the best-known polynomial-time approximation algorithm is the dominating distribution algorithm due to Lund, Phillips and Reingold (FOCS 1994), who showed that the algorithm is $4$-competitive against $\mathrm{OPT}$. We substantially improve their analysis and show that the dominating distribution algorithm is in fact $2$-competitive against $\mathrm{OPT}$. We also show a lower bound of $1.5907$-competitiveness for this algorithm -- to the best of our knowledge, no such lower bound was previously known.

We study learning-augmented streaming algorithms for estimating the value of MAX-CUT in a graph. In the classical streaming model, while a 1/2-approximation for estimating the value of MAX-CUT can be trivially achieved with O(1) words of space, Kapralov and Krachun [STOC’19] showed that this is essentially the best possible: for any ε > 0, any (randomized) single-pass streaming algorithm that achieves an approximation ratio of at least 1/2 + ε requires Ω(n / 2^poly(1/ε)) space. We show that it is possible to surpass the 1/2-approximation barrier using just O(1) words of space by leveraging a (machine learned) oracle. Specifically, we consider streaming algorithms that are equipped with an ε-accurate oracle that for each vertex in the graph, returns its correct label in {-1, +1}, corresponding to an optimal MAX-CUT solution in the graph, with some probability 1/2 + ε, and the incorrect label otherwise. Within this framework, we present a single-pass algorithm that approximates the value of MAX-CUT to within a factor of 1/2 + Ω(ε²) with probability at least 2/3 for insertion-only streams, using only poly(1/ε) words of space. We also extend our algorithm to fully dynamic streams while maintaining a space complexity of poly(1/ε,log n) words.

We study the metric Steiner tree problem in the sublinear query model. In this problem, for a set of $n$ points $V$ in a metric space given to us by means of query access to an $n\times n$ matrix $w$, and a set of terminals $T\subseteq V$, the goal is to find the minimum-weight subset of the edges that connects all the terminal vertices. Recently, Chen, Khanna and Tan [SODA'23] gave an algorithm that uses $\widetilde{O}(n^{13/7})$ queries and outputs a $(2-η)$-estimate of the metric Steiner tree weight, where $η>0$ is a universal constant. A key component in their algorithm is a sublinear algorithm for a particular set cover problem where, given a set system $(U, F)$, the goal is to provide a multiplicative-additive estimate for $|U|-\textsf{SC}(U, F)$. Here $U$ is the set of elements, $F$ is the collection of sets, and $\textsf{SC}(U, F)$ denotes the optimal set cover size of $(U, F)$. In particular, their algorithm returns a $(1/4, \varepsilon\cdot|U|)$-multiplicative-additive estimate for this set cover problem using $\widetilde{O}(|F|^{7/4})$ membership oracle queries (querying whether a set $S$ contains an $e$), where $\varepsilon$ is a fixed constant. In this work, we improve the query complexity of $(2-η)$-estimating the metric Steiner tree weight to $\widetilde{O}(n^{5/3})$ by showing a $(1/2, \varepsilon \cdot |U|)$-estimate for the above set cover problem using $\widetilde{O}(|F|^{5/3})$ membership queries. To design our set cover algorithm, we estimate the size of a random greedy maximal matching for an auxiliary multigraph that the algorithm constructs implicitly, without access to its adjacency list or matrix.

We consider the problem of graph searching with prediction recently introduced by Banerjee et al. (2023). In this problem, an agent starting at some vertex r has to traverse a (potentially unknown) graph G to find a hidden goal node g while minimizing the total distance traveled. We study a setting in which at any node v, the agent receives a noisy estimate of the distance from v to g. We design algorithms for this search task on unknown graphs. We establish the first formal guarantees on unknown weighted graphs and provide lower bounds showing that the algorithms we propose have optimal or nearly-optimal dependence on the prediction error. Further, we perform numerical experiments demonstrating that in addition to being robust to adversarial error, our algorithms perform well in typical instances in which the error is stochastic. Finally, we provide simpler performance bounds on the algorithms of Banerjee et al. (2023) for the case of searching on a known graph and establish new lower bounds for this setting.

In this paper, we study the individual preference (IP) stability, which is an notion capturing individual fairness and stability in clustering. Within this setting, a clustering is $\alpha$-IP stable when each data point’s average distance to its cluster is no more than $\alpha$ times its average distance to any other cluster. In this paper, we study the natural local search algorithm for IP stable clustering. Our analysis confirms a $O(\log n)$-IP stability guarantee for this algorithm, where $n$ denotes the number of points in the input. Furthermore, by refining the local search approach, we show it runs in an almost linear time, $\tilde{O}(nk)$.

We study the k-connectivity augmentation problem (k-CAP) in the single-pass streaming model. Given a (k − 1)-edge connected graph G = (V, E) that is stored in memory, and a stream of weighted edges (also called links) L with weights in {0, 1, . . ., W }, the goal is to choose a minimum weight subset L′ ⊆ L of the links such that G′ = (V, E ∪ L′) is k-edge connected. We give a (2 + ϵ)approximation algorithm for this problem which requires to store O(ϵ−1n log n) words. Moreover, we show the tightness of our result: Any algorithm with better than 2-approximation for the problem requires Ω(n2) bits of space even when k = 2. This establishes a gap between the optimal approximation factor one can obtain in the streaming vs the offline setting for k-CAP. We further consider a natural generalization to the fully streaming model where both E and L arrive in the stream in an arbitrary order. We show that this problem has a space lower bound that matches the best possible size of a spanner of the same approximation ratio. Following this, we give improved results for spanners on weighted graphs: We show a streaming algorithm that finds a (2t − 1 + ϵ)-approximate weighted spanner of size at most O(ϵ−1n1+1/t log n) for integer t, whereas the best prior streaming algorithm for spanner on weighted graphs had size depending on log W. We believe that this result is of independent interest. Using our spanner result, we provide an optimal O(t)-approximation for k-CAP in the fully streaming model with O(nk + n1+1/t) words of space. Finally we apply our results to network design problems such as Steiner tree augmentation problem (STAP), k-edge connected spanning subgraph (k-ECSS) and the general Survivable Network Design problem (SNDP). In particular, we show a single-pass O(t log k)-approximation for SNDP using O(kn1+1/t) words of space, where k is the maximum connectivity requirement.

In strategic classification, agents modify their features, at a cost, to obtain a positive classification outcome from the learner’s classifier, typically assuming agents have full knowledge of the deployed classifier. In contrast, we consider a Bayesian setting where agents have a common distributional prior on the classifier being used and agents manipulate their features to maximize their expected utility according to this prior. The learner can reveal truthful, yet not necessarily complete, information about the classifier to the agents, aiming to release just enough information to shape the agents' behavior and thus maximize accuracy. We show that partial information release can counter-intuitively benefit the learner’s accuracy, allowing qualified agents to pass the classifier while preventing unqualified agents from doing so. Despite the intractability of computing the best response of an agent in the general case, we provide oracle-efficient algorithms for scenarios where the learner’s hypothesis class consists of low-dimensional linear classifiers or when the agents’ cost function satisfies a sub-modularity condition. Additionally, we address the learner’s optimization problem, offering both positive and negative results on determining the optimal information release to maximize expected accuracy, particularly in settings where an agent’s qualification can be represented by a real-valued number.

Low-rank approximation and column subset selection are two fundamental and related problems that are applied across a wealth of machine learning applications. In this paper, we study the question of socially fair low-rank approximation and socially fair column subset selection, where the goal is to minimize the loss over all sub-populations of the data. We show that surprisingly, even constant-factor approximation to fair low-rank approximation requires exponential time under certain standard complexity hypotheses. On the positive side, we give an algorithm for fair low-rank approximation that, for a constant number of groups and constant-factor accuracy, runs in $2^{\text{poly}(k)}$ rather than the naive $n^{\text{poly}(k)}$, which is a substantial improvement when the dataset has a large number $n$ of observations. We then show that there exist bicriteria approximation algorithms for fair low-rank approximation and fair column subset selection that runs in polynomial time.

We provide new approximation algorithms for the Red-Blue Set Cover and Circuit Minimum Monotone Satisfying Assignment (MMSA) problems. Our algorithm for Red-Blue Set Cover achieves Õ(m^{1/3})-approximation improving on the Õ(m^{1/2})-approximation due to Elkin and Peleg (where m is the number of sets). Our approximation algorithm for MMSA_t (for circuits of depth t) gives an Õ(N^{1-δ}) approximation for δ = 1/32^{3-⌈t/2⌉}, where N is the number of gates and variables. No non-trivial approximation algorithms for MMSA_t with t ≥ 4 were previously known. We complement these results with lower bounds for these problems: For Red-Blue Set Cover, we provide a nearly approximation preserving reduction from Min k-Union that gives an ̃Ω(m^{1/4 - ε}) hardness under the Dense-vs-Random conjecture, while for MMSA we sketch a proof that an SDP relaxation strengthened by Sherali-Adams has an integrality gap of N^{1-ε} where ε → 0 as the circuit depth t → ∞.

We initiate the study of strategic behavior in screening processes with multiple classifiers. We focus on two contrasting settings: a "conjunctive” setting in which an individual must satisfy all classifiers simultaneously, and a sequential setting in which an individual to succeed must satisfy classifiers one at a time. In other words, we introduce the combination of strategic classificationwith screening processes. We show that sequential screening pipelines exhibit new and surprising behavior where individuals can exploit the sequential ordering of the tests to "zig-zag” between classifiers without having to simultaneously satisfy all of them. We demonstrate an individual can obtain a positive outcome using a limited manipulation budget even when far from the intersection of the positive regions of every classifier. Finally, we consider a learner whose goal is to design a sequential screening process that is robust to such manipulations, and provide a construction for the learner that optimizes a natural objective.

This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick $k$ centers with the minimum clustering cost such that each group is at least minimally represented in the centers set and no group dominates the centers set. More precisely, given a set of $n$ points in a metric space $(P, d)$ where each point belongs to one of the $\ell$ different demographics (i.e., $P = P_1 \uplus P_2 \uplus \cdots \uplus P_\ell$) and a set of $\ell$ intervals $[\alpha_1, \beta_1], \cdots, [\alpha_\ell, \beta_\ell]$ on desired number of centers from each group, the goal is to pick a set of $k$ centers $C$ with minimum $\ell_p$-clustering cost (i.e., $(\sum_{v\in P} d(v,C)^p)^{1/p}$) such that for each group $i\in \ell$, $|C\cap P_i| \in [\alpha_i, \beta_i]$. In particular, the fair range $\ell_p$-clustering captures fair range $k$-center, $k$-median and $k$-means as its special cases. In this work, we provide an efficient constant factor approximation algorithm for the fair range $\ell_p$-clustering for all values of $p\in [1,\infty)$.

Individual preference (IP) stability, introduced by Ahmadi et al. (ICML 2022), is a natural clustering objective inspired by stability and fairness constraints. A clustering is $\alpha$-IP stable if the average distance of every data point to its own cluster is at most $\alpha$ times the average distance to any other cluster. Unfortunately, determining if a dataset admits a $1$-IP stable clustering is NP-Hard. Moreover, before this work, it was unknown if an $o(n)$-IP stable clustering always exists, as the prior state of the art only guaranteed an $O(n)$-IP stable clustering. We close this gap in understanding and show that an $O(1)$-IP stable clustering always exists for general metrics, and we give an efficient algorithm which outputs such a clustering. We also introduce generalizations of IP stability beyond average distance and give efficient near optimal algorithms in the cases where we consider the maximum and minimum distances within and between clusters.

Estimating frequencies of elements appearing in a data stream is a key task in large-scale data analysis. Popular sketching approaches to this problem (e.g., CountMin and CountSketch) come with worst-case guarantees that probabilistically bound the error of the estimated frequencies for any possible input. The work of Hsu et al.~(2019) introduced the idea of using machine learning to tailor sketching algorithms to the specific data distribution they are being run on. In particular, their learning-augmented frequency estimation algorithm uses a learned heavy-hitter oracle which predicts which elements will appear many times in the stream. We give a novel algorithm, which in some parameter regimes, already theoretically outperforms the learning based algorithm of Hsu et al. *without* the use of any predictions. Augmenting our algorithm with heavy-hitter predictions further reduces the error and improves upon the state of the art. Empirically, our algorithms achieve superior performance in all experiments compared to prior approaches.

Given a set of points of interest, a volumetric spanner is a subset of the points using which all the points can be expressed using "small" coefficients (measured in an appropriate norm). Formally, given a set of vectors $X = [v_1, v_2, \dots, v_n]$, the goal is to find $T \subseteq [n]$ such that every $v \in X$ can be expressed as $\sum_{i\in T} \alpha_i v_i$, with $\Vert \alpha \Vert$ being small. This notion, which has also been referred to as a well-conditioned basis, has found several applications, including bandit linear optimization, determinant maximization, and matrix low rank approximation. In this paper, we give almost optimal bounds on the size of volumetric spanners for all $\ell_p$ norms, and show that they can be constructed using a simple local search procedure. We then show the applications of our result to other tasks and in particular the problem of finding coresets for the Minimum Volume Enclosing Ellipsoid (MVEE) problem.

We consider the $k$-clustering problem with $\ell_p$-norm cost, which includes $k$-median, $k$-means and $k$-center, under an individual notion of fairness proposed by Jung et al. [2020]: given a set of points $P$ of size $n$, a set of $k$ centers induces a fair clustering if every point in $P$ has a center among its $n/k$ closest neighbors. Mahabadi and Vakilian [2020] presented a $( p^{O(p)},7)$-bicriteria approximation for fair clustering with $\ell_p$-norm cost: every point finds a center within distance at most $7$ times its distance to its $(n/k)$-th closest neighbor and the $\ell_p$-norm cost of the solution is at most $p^{O(p)}$ times the cost of an optimal fair solution. In this work, for any $\epsilon>0$, we present an improved $(16^p +\epsilon,3)$-bicriteria for this problem. Moreover, for $p=1$ ($k$-median) and $p=\infty$ ($k$-center), we present improved cost-approximation factors $7.081+\epsilon$ and $3+\epsilon$ respectively. To achieve our guarantees, we extend the framework of [Charikar et al.,2002, Swamy, 2016] and devise a $16^p$-approximation algorithm for the facility location with $\ell_p$-norm cost under matroid constraint which might be of an independent interest. Besides, our approach suggests a reduction from our individually fair clustering to a clustering with a group fairness requirement proposed by [Kleindessner et al. 2019], which is essentially the median matroid problem.

In this paper, we propose a natural notion of individual preference (IP) stability for clustering, which asks that every data point, on average, is closer to the points in its own cluster than to the points in any other cluster. Our notion can be motivated from several perspectives, including game theory and algorithmic fairness. We study several questions related to our proposed notion. We first show that deciding whether a given data set allows for an IP-stable clustering in general is NP-hard. As a result, we explore the design of efficient algorithms for finding IP-stable clusterings in some restricted metric spaces. We present a polytime algorithm to find a clustering satisfying exact IP-stability on the real line, and an efficient algorithm to find an IP-stable 2-clustering for a tree metric. We also consider relaxing the stability constraint, i.e., every data point should not be too far from its own cluster compared to any other cluster. For this case, we provide polytime algorithms with different guarantees. We evaluate some of our algorithms and several standard clustering approaches on real data sets.

We consider the question of speeding up classic graph algorithms with machine-learned predictions. In this model, algorithms are furnished with extra advice learned from past or similar instances. Given the additional information, we aim to improve upon the traditional worst-case run-time guarantees. Our contributions are the following: (i) We give a faster algorithm for minimum-weight bipartite matching via learned duals, improving the recent result by Dinitz, Im, Lavastida, Moseley and Vassilvitskii (NeurIPS, 2021); (ii) We extend the learned dual approach to the single-source shortest path problem (with negative edge lengths), achieving an almost linear runtime given sufficiently accurate predictions which improves upon the classic fastest algorithm due to Goldberg (SIAM J. Comput., 1995); (iii) We provide a general reduction-based framework for learning-based graph algorithms, leading to new algorithms for degree-constrained subgraph and minimum-cost 0-1 flow, based on reductions to bipartite matching and the shortest path problem. Finally, we give a set of general learnability theorems, showing that the predictions required by our algorithms can be efficiently learned in a PAC fashion.

We introduce the (p, q)-Fair Clustering problem. In this problem, we are given a set of points P and a collection of different weight functions W. We would like to find a clustering which minimizes the ℓq-norm of the vector over W of the ℓp-norms of the weighted distances of points in P from the centers. This generalizes various clustering problems, including Socially Fair k-Median and k-Means, and is closely connected to other problems such as Densest k-Subgraph and Min k-Union. We utilize convex programming techniques to approximate the (p, q)-Fair Clustering problem for different values of p and q. When p ≥ q, we get an O(k(p–q)/(2pq)), which nearly matches a kΩ((p–q)/(pq)) lower bound based on conjectured hardness of Min k-Union and other problems. When q ≥ p, we get an approximation which is independent of the size of the input for bounded p,q, and also matches the recent O((log n/(log log n))1/p)-approximation for (p, ∞)-Fair Clustering by Makarychev and Vakilian (COLT 2021).

We present an (e^{O(p)} (log \ell) / (log log \ell))-approximation algorithm for socially fair clustering with the l_p-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of \ell groups. The goal is to find a k-medians, k-means, or, more generally, l_p-clustering that is simultaneously good for all of the groups. More precisely, we need to find a set of k centers C so as to minimize the maximum over all groups j of \sum_{u in group j} d(u, C)^p. The socially fair clustering problem was independently proposed by Abbasi, Bhaskara, and Venkatasubramanian (2021) and Ghadiri, Samadi, and Vempala (2021). Our algorithm improves and generalizes their O(\ell)-approximation algorithms for the problem. The natural LP relaxation for the problem has an integrality gap of \Omega(\ell). In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of \Theta((log \ell) / (log log \ell)) for a fixed p. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. (2021).

We study the problem of designing online algorithms given advice about the input. While prior work had focused on deterministic advice, we only assume distributional access to the instances of interest, and the goal is to learn a competitive algorithm given access to i.i.d. samples. We aim to be competitive against an adversary with prior knowledge of the distribution, while also performing well against worst-case inputs. We focus on the classical online problems of ski-rental and prophet-inequalities, and provide sample complexity bounds for the underlying learning tasks. First, we point out that for general distributions it is information-theoretically impossible to beat the worst-case competitive-ratio with any finite sample size. As our main contribution, we establish strong positive results for well-behaved distributions. Specifically, for the broad class of log-concave distributions, we show that $\mathrm{poly}(1/\epsilon)$ samples suffice to obtain $(1+\epsilon)$-competitive ratio. Finally, we show that this sample upper bound is close to best possible, even for very simple classes of distributions.

We consider node-weighted survivable network design (SNDP) in planar graphs and minor-closed families of graphs. The input consists of a node-weighted undirected graph G = ( V , E ) and integer connectivity requirements r ( uv ) for each unordered pair of nodes uv . The goal is to find a minimum weighted subgraph H of G such that H contains r ( uv ) disjoint paths between u and v for each node pair uv . Three versions of the problem are edge-connectivity SNDP (EC-SNDP), element-connectivity SNDP (Elem-SNDP), and vertex-connectivity SNDP (VC-SNDP), depending on whether the paths are required to be edge, element, or vertex disjoint, respectively. Our main result is an O ( k )-approximation algorithm for EC-SNDP and Elem-SNDP when the input graph is planar or more generally if it belongs to a proper minor-closed family of graphs; here, k = max uv r ( uv ) is the maximum connectivity requirement. This improves upon the O ( k log n )-approximation known for node-weighted EC-SNDP and Elem-SNDP in general graphs [31]. We also obtain an O (1) approximation for node-weighted VC-SNDP when the connectivity requirements are in {0, 1, 2}; for higher connectivity our result for Elem-SNDP can be used in a black-box fashion to obtain a logarithmic factor improvement over currently known general graph results. Our results are inspired by, and generalize, the work of Demaine, Hajiaghayi, and Klein [13], who obtained constant factor approximations for node-weighted Steiner tree and Steiner forest problems in planar graphs and proper minor-closed families of graphs via a primal-dual algorithm.

We give a local search based algorithm for $k$-median and $k$-means (and more generally for any $k$-clustering with $\ell_p$ norm cost function) from the perspective of individual fairness. More precisely, for a point $x$ in a point set $P$ of size $n$, let $r(x)$ be the minimum radius such that the ball of radius $r(x)$ centered at $x$ has at least $n/k$ points from $P$. Intuitively, if a set of $k$ random points are chosen from $P$ as centers, every point $x\in P$ expects to have a center within radius $r(x)$. In this work, we show how to get an approximately optimal such fair $k$-clustering: The $k$-median ($k$-means) cost of our solution is within a constant factor of the cost of an optimal fair $k$-clustering, and our solution approximately satisfies the fairness condition (also within a constant factor).

We design a Local Computation Algorithm (LCA) for the set cover problem. Given a set system where each set has size at most s and each element is contained in at most t sets, the algorithm reports whether a given set is in some fixed set cover whose expected size is O(log s) times the minimum fractional set cover value. Our algorithm requires sO(log s) tO(log s+log log t)) queries. This result improves upon the application of the reduction of [Parnas and Ron, TCS’07] on the result of [Kuhn et al., SODA’06], which leads to a query complexity of (st) O(log s · log t). To obtain this result, we design a parallel set cover algorithm that admits an efficient simulation in the LCA model by using a sparsification technique introduced in [Ghaffari and Uitto, SODA’19] for the maximal independent set problem. The parallel algorithm adds a random subset of the sets to the solution in a style similar to the PRAM algorithm of [Berger et al., FOCS’89]. However, our algorithm differs in the way that it never revokes its decisions, which results in a fewer number of adaptive rounds. This requires a novel approximation analysis which might be of independent interest.

A distance matrix $A \in \mathbb{R}^{n \times m}$ represents all pairwise distances, $A_{i,j} = d(x_i,y_j)$, between two point sets $x_1,\dotsc,x_n$ and $y_1,\dotsc,y_m$ in an arbitrary metric space $(\mathcal{Z},d)$. Such matrices arise in various computational contexts such as learning image manifolds, handwriting recognition, and multi-dimensional unfolding. In this work we study algorithms for low-rank approximation of distance matrices. Recent work by Bakshi and Woodruff (NeurIPS 2018) showed it is possible to compute a rank-$k$ approximation of a distance matrix in time $O((n+m)^{1+\gamma}) \mathrm{poly}(k,1/\epsilon)$, where $\epsilon>0$ is an error parameter and $\gamma>0$ is an arbitrarily small constant. Notably, their bound is sublinear in the matrix size, which is unachieveable for general matrices. We present an algorithm that is both simpler and more efficient. It reads only $O((n+m)k/\epsilon)$ entries of the input matrix, and has a running time of $O(n+m) \cdot \mathrm{poly}(k,1/\epsilon)$. We complement the sample complexity of our algorithm with a matching lower bound on the number of entries that must be ready by any algorithm. We provide experimental results to validate the approximation quality and running time of our algorithm

We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC).

We study the fair variant of the classic k-median problem introduced by (Chierichetti et al., NeurIPS 2017) in which the points are colored, and the goal is to minimize the same average distance objective as in the standard $k$-median problem while ensuring that all clusters have an “approximately equal” number of points of each color. (Chierichetti et al., NeurIPS 2017) proposed a two-phase algorithm for fair $k$-clustering. In the first step, the pointset is partitioned into subsets called fairlets that satisfy the fairness requirement and approximately preserve the k-median objective. In the second step, fairlets are merged into k clusters by one of the existing k-median algorithms. The running time of this algorithm is dominated by the first step, which takes super-quadratic time. In this paper, we present a practical approximate fairlet decomposition algorithm that runs in nearly linear time.

A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the best-known existential size-stretch trade-off efficiently. Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a single processor to store. To tackle this challenge, we initiate the study of local computation algorithms (LCAs) for graph spanners in general graphs, where the algorithm should locally decide whether a given edge $(u,v) \in E$ belongs to the output spanner. Such LCAs give the user the `illusion' that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present the following results: -For general $n$-vertex graphs and $r \in \{2,3\}$, there exists an LCA for $(2r-1)$-spanners with $\widetilde{O}(n^{1+1/r})$ edges and sublinear probe complexity of $\widetilde{O}(n^{1-1/2r})$. These size/stretch tradeoffs are best possible (up to polylogarithmic factors). -For every $k \geq 1$ and $n$-vertex graph with maximum degree $Δ$, there exists an LCA for $O(k^2)$ spanners with $\widetilde{O}(n^{1+1/k})$ edges, probe complexity of $\widetilde{O}(Δ^4 n^{2/3})$, and random seed of size $\mathrm{polylog}(n)$. This improves upon, and extends the work of [Lenzen-Levi, 2018]. We also complement our results by providing a polynomial lower bound on the probe complexity of LCAs for graph spanners that holds even for the simpler task of computing a sparse connected subgraph with $o(m)$ edges.

We introduce a “learning-based” algorithm for the low-rank decomposition problem: given an $n \times d$ matrix $A$, and a parameter $k$, compute a rank-$k$ matrix $A'$ that minimizes the approximation loss $\|A-A'\|_F$. The algorithm uses a training set of input matrices in order to optimize its performance. Specifically, some of the most efficient approximate algorithms for computing low-rank approximations proceed by computing a projection $SA$, where $S$ is a sparse random $m \times n$ “sketching matrix”, and then performing the singular value decomposition of $SA$. We show how to replace the random matrix $S$ with a “learned” matrix of the same sparsity to reduce the error. Our experiments show that, for multiple types of data sets, a learned sketch matrix can substantially reduce the approximation loss compared to a random matrix $S$, sometimes up to one order of magnitude. We also study mixed matrices where only some of the rows are trained and the remaining ones are random, and show that matrices still offer improved performance while retaining worst-case guarantees. Finally, to understand the theoretical aspects of our approach, we study the special case of $m=1$. In particular, we give an approximation algorithm for minimizing the empirical loss, with approximation factor depending on the stable rank of matrices in the training set. We also show generalization bounds for the sketch matrix learning problem.

We study the maximum k-coverage problem in the general edge-arrival streaming model: given a collection of m sets F, each subset of a ground set of elements U of size n, the task is to find k sets whose coverage is maximized. The sets are specified as a sequence of (element, set) pairs in an arbitrary order. Our main result is a tight (up to polylogarithmic factors) trade-off between the space complexity and the approximation factor α\in(1/(1-1/e), \tildeOmega (\sqrtm )]$ of any single-pass streaming algorithm that estimates the maximum coverage size. Specifically, we show that the optimal space bound is $\tildeTheta (m/α^2)$. Moreover, we design a single-pass algorithm that reports an α-approximate solution in $\tildeO (m/α^2 + k)$ space. Our algorithm heavily exploits data stream sketching techniques, which could lead to further connections between vector sketching methods and streaming algorithms for combinatorial optimization tasks.

We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in [17] to the sub-linear query model, that returns an α-approximate cover using Õ(m(n/k)1/(α–1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, the required number of queries is , even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require Ω(nk) queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter k. On the other hand, we show that this bound is not optimal for larger values of the parameter k, as there exists a (1 + ε)-approximation algorithm with Õ(mn/kε2) queries. We show that this bound is essentially tight for sufficiently small constant ε, by establishing a lower bound of query complexity. Our lower-bound results follow by carefully designing two distributions of instances that are hard to distinguish. In particular, our first lower bound involves a probabilistic construction of a certain set system with a minimum set cover of size αk, with the key property that a small number of “almost uniformly distributed” modifications can reduce the minimum set cover size down to k. Thus, these modifications are not detectable unless a large number of queries are asked. We believe that our probabilistic construction technique might find applications to lower bounds for other combinatorial optimization problems.

We study the Fractional Set Cover problem in the streaming model. That is, we consider the relaxation of the set cover problem over a universe of n elements and a collection of m sets, where each set can be picked fractionally, with a value in [0,1]. We present a randomized (1+a)-approximation algorithm that makes p passes over the data, and uses O(polylog(m,n,1/a) (mn^(O(1/(pa)))+n)) memory space. The algorithm works in both the set arrival and the edge arrival models. To the best of our knowledge, this is the first streaming result for the fractional set cover problem. We obtain our results by employing the multiplicative weights update framework in the streaming settings.

It is known that annotating entities in unstructured and semistructured datasets by their concepts improves the effectiveness of answering queries over these datasets. Ideally, one would like to annotate entities of all relevant concepts in a dataset. However, it takes substantial time and computational resources to annotate concepts in large datasets and an organization may have sufficient resources to annotate only a subset of relevant concepts. Clearly, it would like to annotate a subset of concepts that provides the most effective answers to queries over the dataset. We propose a formal framework that quantifies the amount by which annotating entities of concepts from a taxonomy in a dataset improves the effectiveness of answering queries over the dataset. Because the problem is NP-hard, we propose an efficient approximation for the problem. Our extensive empirical studies validate our framework and show the accuracy and efficiency of our algorithm.

We consider the classic Set Cover problem in the data stream model. For n elements and m sets (m ≥ n) we give a O(1/δ)-pass algorithm with a strongly sub-linear ~O(mnδ) space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [10] that uses exponentially larger number of passes. We complement this result by showing that the tradeoff between the number of passes and space exhibited by our algorithm is tight, at least when the approximation factor is equal to 1. Specifically, we show that any algorithm that computes set cover exactly using ({1 over 2δ}-1) passes must use ~Ω(mnδ) space in the regime of m=O(n). Furthermore, we consider the problem in the geometric setting where the elements are points in R2 and sets are either discs, axis-parallel rectangles, or fat triangles in the plane, and show that our algorithm (with a slight modification) uses the optimal ~O(n) space to find a logarithmic approximation in O(1/δ) passes.

It is well established that extracting and annotating occurrences of entities in a collection of unstructured text documents with their concepts improves the effectiveness of answering queries over the collection. However, it is very resource intensive to create and maintain large annotated collections. Since the available resources of an enterprise are limited and/or its users may have urgent information needs, it may have to select only a subset of relevant concepts for extraction and annotation. We call this subset a conceptual design for the annotated collection. In this article, we introduce and formally define the problem of cost-effective conceptual design where, given a collection, a set of relevant concepts, and a fixed budget, one likes to find a conceptual design that most improves the effectiveness of answering queries over the collection. We provide efficient algorithms for special cases of the problem and prove it is generally NP-hard in the number of relevant concepts. We propose three efficient approximations to solve the problem: a greedy algorithm, an approximate popularity maximization (APM for short), and approximate annotation-benefit maximization (AAM for short). We show that, if there are no constraints regrading the overlap of concepts, APM is a fully polynomial time approximation scheme. We also prove that if the relevant concepts are mutually exclusive, the greedy algorithm delivers a constant approximation ratio if the concepts are equally costly, APM has a constant approximation ratio, and AAM is a fully polynomial-time approximation scheme. Our empirical results using a Wikipedia collection and a search engine query log validate the proposed formalization of the problem and show that APM and AAM efficiently compute conceptual designs. They also indicate that, in general, APM delivers the optimal conceptual designs if the relevant concepts are not mutually exclusive. Also, if the relevant concepts are mutually exclusive, the conceptual designs delivered by AAM improve the effectiveness of answering queries over the collection more than the solutions provided by APM.

It is well established that extracting and annotating occurrences of entities in a collection of unstructured text documents with their concepts improve the effectiveness of answering queries over the collection. However, it is very resource intensive to create and maintain large annotated collections. Since the available resources of an enterprise are limited and/or its users may have urgent information needs, it may have to select only a subset of relevant concepts for extraction and annotation. We call this subset a conceptual design for the annotated collection. In this paper, we introduce the problem of cost effective conceptual design, where given a collection, a set of relevant concepts, and a fixed budget, one likes to find a conceptual design that improves the effectiveness of answering queries over the collection the most. We prove that the problem is generally NP-hard in the number of relevant concepts and propose two efficient approximation algorithms to solve the problem: Approximate Popularity Maximization (APM for short) and Approximate Annotation-benefit Maximization (AAM for short). We show that if there is not any constraints regrading the overlap of concepts, APM is a fully polynomial time approximation scheme. We also prove that if the relevant concepts are mutually exclusive, APM has a constant approximation ratio and AAM is a fully polynomial time approximation scheme. Our empirical results using Wikipedia collection and a search engine query log validate the proposed formalization of the problem and show that APM and AAM efficiently compute conceptual designs. They also indicate that in general APM delivers the optimal conceptual designs if the relevant concepts are not mutually exclusive. Also, if the relevant concepts are mutually exclusive, the conceptual designs delivered by AAM improve the effectiveness of answering queries over the collection more than the solutions provided by APM.

We consider degree bounded network design problems with element and vertex connectivity requirements. In the degree bounded Survivable Network Design (SNDP) problem, the input is an undirected graph G = (V, E) with weights w(e) on the edges and degree bounds b(v) on the vertices, and connectivity requirements r(uv) for each pair uv of vertices. The goal is to select a minimum-weight subgraph H of G that meets the connectivity requirements and it satisfies the degree bounds on the vertices: for each pair uv of vertices, H has r(uv) disjoint paths between u and v; additionally, each vertex v is incident to at most b(v) edges in H. We give the first (O(1), O(1) · b(v)) bicriteria approximation algorithms for the degree-bounded SNDP problem with element connectivity requirements and for several degree-bounded SNDP problems with vertex connectivity requirements. Our algorithms construct a subgraph H whose weight is at most O(1) times the optimal such that each vertex v is incident to at most O(1) · b(v) edges in H. We can also extend our approach to network design problems in directed graphs with out-degree constraints to obtain (O(1), O(1) · b+(v)) bicriteria approximation.