Blockchains have popularized automated market makers (AMMs), applications that run on a blockchain, maintain a pool of crypto-assets, and execute trades with users governed by some pricing function. AMMs have also introduced a significant challenge known as the Miner Extractable Value (MEV). Specifically, miners who control the contents and sequencing of transactions in a block can extract value by front-running and back-running users' transactions, creating arbitrage opportunities that guarantee them risk-free returns. MEV not only harms ordinary users, but more critically, encourages miners to auction off favorable transaction placements to users and arbitragers. This has fostered a more centralized off-chain eco-system, departing from the decentralized equilibrium originally envisioned for the blockchain infrastructure layer. In this paper, we consider how to design AMM mechanisms that eliminate MEV opportunities. Specifically, we propose a new AMM mechanism that processes all transactions contained within a block according to some pre-defined rules, ensuring that some constant potential function is maintained after processing the batch. We show that our new mechanism satisfies two tiers of guarantees. First, for legacy blockchains where each block is proposed by a single (possibly rotating) miner, we prove that our mechanism satisfies arbitrage resilience, i.e., a miner cannot gain risk-free profit. Second, for blockchains where the block proposal process is decentralized and offers sequencing-fairness, we prove a strictly stronger notion called strategy proofness - roughly speaking, we guarantee that any individual user’s best response is to follow the honest strategy. Our results complement prior works on MEV resilience in the following senses. First, prior works have shown impossibilities to address MEV entirely at the consensus level. Our work demonstrates a new paradigm of mechanism design at the application (i.e., smart contract) layer to ensure provable guarantees of strategy proofness. Second, many works have attempted to augment the underlying consensus protocol with extra properties such as sequencing fairness. While most previous works heuristically argued why these extra properties help to mitigate MEV, our work demonstrates in a mathematically formal manner how to leverage such consensus-level properties to aid the design of strategy-proof mechanisms.

We give online algorithms for $k$-Means(more generally, $(k, z)$-Clustering) with nearly optimal consistency (a notion suggested by Lattanzi & Vassilvitskii (2017)). Our result turns any $\alpha$-approximate offline algorithm for clustering into an $(1+\epsilon)\alpha^2$-competitive online algorithm for clustering with $O(k \text{poly} \log n)$ consistency. This consistency bound is optimal up to $\text{poly} \log(n)$ factors. Plugging in the offline algorithm that returns the exact optimal solution, we obtain the first $(1 + \epsilon)$-competitive online algorithm for clustering that achieves a linear in $k$ consistency. This simultaneously improves several previous results (Lattanzi & Vassilvitskii, 2017; Fichtenberger et al., 2021). We validate the performance of our algorithm on real datasets by plugging in the practically efficient $k$-Means++ algorithm. Our online algorithm makes $k$-Means++ achieve good consistency with little overhead to the quality of solutions.

We investigate the closest distribution refinements problem, which involves a vertex-weighted bipartite graph as input, where the vertex weights on each side sum to 1 and represent a probability distribution. A refinement of one side’s distribution is an edge distribution that corresponds to distributing the weight of each vertex from that side to its incident edges. The objective is to identify a pair of distribution refinements for both sides of the bipartite graph such that the two edge distributions are as close as possible with respect to a specific divergence notion. This problem is a generalization of transportation, in which the special case occurs when the two closest distributions are identical. The problem has recently emerged in the context of composing differentially oblivious mechanisms. Our main result demonstrates that a universal refinement pair exists, which is simultaneously closest under all divergence notions that satisfy the data processing inequality. Since differential obliviousness can be examined using various divergence notions, such a universally closest refinement pair offers a powerful tool in relation to such applications. We discover that this pair can be achieved via locally verifiable optimality conditions. Specifically, we observe that it is equivalent to the following problems, which have been traditionally studied in distinct research communities: (1) hypergraph density decomposition, and (2) symmetric Fisher Market equilibrium. We adopt a symmetric perspective of hypergraph density decomposition, in which hyperedges and nodes play equivalent roles. This symmetric decomposition serves as a tool for deriving precise characterizations of optimal solutions for other problems and enables the application of algorithms from one problem to another. This connection allows existing algorithms for computing or approximating the Fisher market equilibrium to be adapted for all the aforementioned problems. For example, this approach allows the well-known iterative proportional response process to provide approximations for the corresponding problems with multiplicative error in distributed settings, whereas previously, only absolute error had been achieved in these contexts. Our study contributes to the understanding of various problems within a unified framework, which may serve as a foundation for connecting other problems in the future.

Graphs are crucial for modeling complex relationships in fields like social networks and biology. A key aspect in graph theory is identifying dense subgraphs, with applications in various domains. Density decomposition refines this by analyzing a graph's global density structure. This concept has been independently rediscovered in research areas like graph mining, algorithm design, and economics. Recent advancements in maximum-flow algorithms allow for nearly-linear time computation of the density vector, but they struggle with large real-world graphs. To address this, iterative first-order methods based on convex optimization, such as the Frank-Wolfe algorithm and momentum-based methods like accelerated FISTA, have been developed to approximate density vectors. This work explores density decomposition through market dynamics, where edges represent buyers and nodes represent sellers in a Fisher market model. The iterative proportional response process, which converges to the Fisher market equilibrium, provides an alternative interpretation of density decomposition. In each step, agents allocate resources based on the proportion of benefit received, moving the system toward equilibrium. Since each proportional response update can be seen as a gradient descent step, we investigate whether momentum methods can speed up convergence. Traditional momentum uses a linear combination of current and previous solutions, whereas our novel exponential momentum variant uses geometric interpolation, aligning better with proportional adjustments. Empirical evaluations on large-scale real-world and synthetic graphs confirm the effectiveness of our methods. Notably, the proportional response algorithm with exponential momentum outperforms existing methods, delivering improvements by several orders of magnitude in some cases. This advancement is significant, as the resulting density vector is crucial for many downstream tasks in graph mining and algorithm design.

We examine a private ADMM variant for (strongly) convex objectives which is a primal-dual iterative method. Each iteration has a user with a private function used to update the primal variable, masked by Gaussian noise for local privacy, without directly adding noise to the dual variable. Privacy amplification by iteration explores if noises from later iterations can enhance the privacy guarantee when releasing final variables after the last iteration. Cyffers et al. explored privacy amplification by iteration for the proximal ADMM variant, where a user's entire private function is accessed and noise is added to the primal variable. In contrast, we examine a private ADMM variant requiring just one gradient access to a user's function, but both primal and dual variables must be passed between successive iterations. To apply Balle et al.'s coupling framework to the gradient ADMM variant, we tackle technical challenges with novel ideas. First, we address the non-expansive mapping issue in ADMM iterations by using a customized norm. Second, because the dual variables are not masked with any noise directly, their privacy guarantees are achieved by treating two consecutive noisy ADMM iterations as a Markov operator. Our main result is that the privacy guarantee for the gradient ADMM variant can be amplified proportionally to the number of iterations. For strongly convex objective functions, this amplification exponentially increases with the number of iterations. These amplification results align with the previously studied special case of stochastic gradient descent.

Differential obliviousness (DO) is a privacy notion which mandates that the access patterns of a program satisfy differential privacy. Earlier works have shown that in numerous applications, differential obliviousness allows us to circumvent fundamental barriers pertaining to fully oblivious algorithms, resulting in asymptotical (and sometimes even polynomial) performance improvements. Although DO has been applied to various contexts, including the design of algorithms, data structures, and protocols, its compositional properties are not explored until the recent work of Zhou et al. (Eurocrypt'23). Specifically, Zhou et al. showed that the original DO notion is not composable. They then proposed a refinement of DO called neighbor-preserving differential obliviousness (NPDO), and proved a basic composition for NPDO. In Zhou et al.'s basic composition theorem for NPDO, the privacy loss is linear in k for k-fold composition. In comparison, for standard differential privacy, we can enjoy roughly √k loss for k-fold composition by applying the well-known advanced composition theorem given an appropriate parameter range. Therefore, a natural question left open by their work is whether we can also prove an analogous advanced composition for NPDO. In this paper, we answer this question affirmatively. As a key step in proving an advanced composition theorem for NPDO, we define a more operational notion called symmetric NPDO which we prove to be equivalent to NPDO. Using symmetric NPDO as a stepping stone, we also show how to generalize NPDO to more general notions of divergence, resulting in Rényi-NPDO, zeroconcentrated-NPDO, Gassian-NPDO, and g-NPDO notions. We also prove composition theorems for these generalized notions of NPDO.

Given a graph $G$ and a positive integer $k$, the Graphlet Sampling problem asks to sample a connected induced $k$-vertex subgraph of $G$ uniformly at random. Graphlet sampling enhances machine learning applications by transforming graph structures into feature vectors for tasks such as graph classification and subgraph identification, boosting neural network performance, and supporting clustered federated learning by capturing local structures and relationships. A recent work has shown that the problem admits an algorithm that preprocesses $G$ in time $O(nk^2 \log k + m)$, and draws one sample in expected time $k^{O(k)} \log n$, where $n=|V(G)|$ and $m=|E(G)|$. Such an algorithm relies on the assumption that the input graph fits into main memory and it does not seem to be straightforward to adapt it to very large graphs. We consider Graphlet Sampling in the semi-streaming setting, where we have a memory of $M = \Omega(n \log n)$ words, and $G$ can be only read through sequential passes over the edge list. We develop a semi-streaming algorithm that preprocesses $G$ in $p={O}(\log n)$ passes and samples $\Theta(M k^{-O(k)})$ independent uniform $k$-graphlets in $O(k)$ passes. For constant $k$, both phases run in time $O((n+m)\log n)$. We also show that the tradeoff between memory and number of passes of our algorithms is near-optimal. Our extensive evaluation on very large graphs shows the effectiveness of our algorithms.

Finding dense subgraphs in large (hyper)graphs is a key primitive in a variety of real-world application domains, encompassing social network analytics, event detection, biology, and finance. In most such applications, one typically aims at finding several (possibly overlapping) dense subgraphs, which might correspond to communities in social networks or interesting events. While a large amount of work is devoted to finding a single densest subgraph, perhaps surprisingly, the problem of finding several dense subgraphs in weighted hypergraphs with limited overlap has not been studied in a principled way, to the best of our knowledge. In this work, we define and study a natural generalization of the densest subgraph problem in weighted hypergraphs, where the main goal is to find at most k subgraphs with maximum total aggregate density, while satisfying an upper bound on the pairwise weighted Jaccard coefficient, i.e., the ratio of weights of intersection divided by weights of union on two nodes sets of the subgraphs. After showing that such a problem is NP-Hard, we devise an efficient algorithm that comes with provable guarantees in some cases of interest, as well as, an efficient practical heuristic. Our extensive evaluation on large real-world hypergraphs confirms the efficiency and effectiveness of our algorithms.

Vector mean estimation is a central primitive in federated analytics. In vector mean estimation, each user $i \in[n]$ holds a real-valued vector $v_{i} \in[-1,1]^{d}$, and a server wants to estimate the mean of all n vectors; we would additionally like to protect each user’s privacy. In this paper, we consider the k-sparse version of the vector mean estimation problem. That is, suppose each user’s vector has at most k non-zero coordinates in its d-dimensional vector, and moreover, $k \ll d$. In practice, since the universe size d can be very large (e.g., the space of all possible URLs), we would like the per-user communication to be succinct, i.e., independent of or (poly-)logarithmic in the universe size.In this paper, we show matching upper- and lower-bounds for the k-sparse vector mean estimation problem under local differential privacy (LDP). Specifically, we construct new mechanisms that achieve asymptotically optimal error as well as succinct communication, either under user-level-LDP or event-level-LDP. We implement our algorithms and evaluate them on synthetic and real-world datasets. Our experiments show that we can often achieve one or two orders of magnitude reduction in error compared with prior work under typical choices of parameters, while incurring insignificant communication cost.

It is well-known that a program’s memory access pattern can leak information about its input. To thwart such leakage, most existing works adopt the technique of oblivious RAM (ORAM) simulation. Such an obliviousness notion has stimulated much debate. Although ORAM techniques have significantly improved over the past few years, the concrete overheads are arguably still undesirable for real-world systems — part of this overhead is in fact inherent due to a well-known logarithmic ORAM lower bound by Goldreich and Ostrovsky. To make matters worse, when the program’s runtime or output length depend on secret inputs, it may be necessary to perform worst-case padding to achieve full obliviousness and thus incur possibly super-linear overheads. Inspired by the elegant notion of differential privacy, we initiate the study of a new notion of access pattern privacy, which we call “ (ϵ , δ) -differential obliviousness”. We separate the notion of (ϵ , δ) -differential obliviousness from classical obliviousness by considering several fundamental algorithmic abstractions including sorting small-length keys, merging two sorted lists, and range query data structures (akin to binary search trees). We show that by adopting differential obliviousness with reasonable choices of ϵ and δ , not only can one circumvent several impossibilities pertaining to full obliviousness, one can also, in several cases, obtain meaningful privacy with little overhead relative to the non-private baselines (i.e., having privacy “with little extra overhead”). On the other hand, we show that for very demanding choices of ϵ and δ , the same lower bounds for oblivious algorithms would be preserved for (ϵ, δ) -differential obliviousness.

A long line of work in social psychology has studied variations in people’s susceptibility to persuasion—the extent to which they are willing to modify their opinions on a topic. This body of literature suggests an interesting perspective on theoretical models of opinion formation by interacting parties in a network: in addition to considering interventions that directly modify people’s intrinsic opinions, it is also natural to consider interventions that modify people’s susceptibility to persuasion. In this work, motivated by this fact, we propose an influence optimization problem. Specifically, we adopt a popular model for social opinion dynamics, where each agent has some fixed innate opinion, and a resistance that measures the importance it places on its innate opinion; agents influence one another’s opinions through an iterative process. Under certain conditions, this iterative process converges to some equilibrium opinion vector. For the unbudgeted variant of the problem, the goal is to modify the resistance of any number of agents (within some given range) such that the sum of the equilibrium opinions is minimized; for the budgeted variant, in addition the algorithm is given upfront a restriction on the number of agents whose resistance may be modified. We prove that the objective function is in general non-convex. Hence, formulating the problem as a convex program as in an early version of this work (Abebe et al., KDD’18) might have potential correctness issues. We instead analyze the structure of the objective function, and show that any local optimum is also a global optimum, which is somehow surprising as the objective function might not be convex. Furthermore, we combine the iterative process and the local search paradigm to design very efficient algorithms that can solve the unbudgeted variant of the problem optimally on large-scale graphs containing millions of nodes. Finally, we propose and evaluate experimentally a family of heuristics for the budgeted variant of the problem.

Static and dynamic clustering algorithms are a fundamental tool in any machine learning library. Most of the efforts in developing dynamic machine learning and data mining algorithms have been focusing on the sliding window model or more simplistic models. However, in many real-world applications one might need to deal with arbitrary deletions and insertions. For example, one might need to remove data items that are not necessarily the oldest ones, because they have been flagged as containing inappropriate content or due to privacy concerns. Clustering trajectory data might also require to deal with more general update operations. We develop a$(2+\epsilon)$-approximation algorithm for the$k$-center clustering problem with “small” amortized cost under the fully dynamic adversarial model. In such a model, points can be added or removed arbitrarily, provided that the adversary does not have access to the random choices of our algorithm. The amortized cost of our algorithm is poly-logarithmic when the ratio between the maximum and minimum distance between any two points in input is bounded by a polynomial, while$k$and$\epsilon$are constant. Furthermore, we significantly improve the memory requirement of our fully dynamic algorithm, although at the cost of a worse approximation ratio of$4 +\epsilon$. Our theoretical results are complemented with an extensive experimental evaluation on dynamic data from Twitter, Flickr, as well as trajectory data, demonstrating the effectiveness of our approach.

Massively Parallel Computation (MPC) is a model of computation widely believed to best capture realistic parallel computing architectures such as large-scale MapReduce and Hadoop clusters. Motivated by the fact that many data analytics tasks performed on these platforms involve sensitive user data, we initiate the theoretical exploration of how to leverage MPC architectures to enable efficient, privacy-preserving computation over massive data. Clearly if a computation task does not lend itself to an efficient implementation on MPC even without security, then we cannot hope to compute it efficiently on MPC with security. We show, on the other hand, that any task that can be efficiently computed on MPC can also be securely computed with comparable efficiency. Specifically, we show the following results: - any MPC algorithm can be compiled to a communication-oblivious counterpart while asymptotically preserving its round and space complexity, where communication-obliviousness ensures that any network intermediary observing the communication patterns learn no information about the secret inputs; - assuming the existence of Fully Homomorphic Encryption with a suitable notion of compactness and other standard cryptographic assumptions, any MPC algorithm can be compiled to a secure counterpart that defends against an adversary who controls not only intermediate network routers but additionally up to 1/3 - η fraction of machines (for an arbitrarily small constant η) - moreover, this compilation preserves the round complexity tightly, and preserves the space complexity upto a multiplicative security parameter related blowup. As an initial exploration of this important direction, our work suggests new definitions and proposes novel protocols that blend algorithmic and cryptographic techniques.

We revisit the problem of low-memory robust simulation of interactive protocols over noisy channels. Haeupler [FOCS 2014] considered robust simulation of two-party interactive protocols over oblivious, as well as adaptive, noisy channels. Since the simulation does not need to have fixed communication pattern, the achieved communication rates can circumvent the lower bound proved by Kol and Raz [STOC 2013]. However, a drawback of this approach is that each party needs to remember the whole history of the simulated transcript. In a subsequent manuscript, Haeupler and Resch considered low-memory simulation. The idea was to view the original protocol as a computational DAG and only the identities of the nodes are saved (as opposed to the whole transcript history) for backtracking to reduce memory usage. In this paper, we consider low-memory robust simulation of more general client-server interactive protocols, in which a leader communicates with other members/servers, who do not communicate among themselves; this setting can be applied to information-theoretic multi-server Private Information Retrieval (PIR) schemes. We propose an information-theoretic technique that converts any correct PIR protocol that assumes reliable channels, into a protocol which is both correct and private in the presence of a noisy channel while keeping the space complexity to a minimum. Despite the huge attention that PIR protocols have received in the literature, the existing works assume that the parties communicate using noiseless channels. Moreover, we observe that the approach of Haeupler and Resch to just save the nodes in the aforementioned DAG without taking the transcript history into account will lead to a correctness issue even for oblivious corruptions. We resolve this issue by saving hashes of prefixes of past transcripts. Departing from the DAG representation also allows us to accommodate scenarios where a party can simulate its part of the protocol without any extra knowledge (such as the DAG representation of the whole protocol). In the the two-party setting, our simulation has the same dependence on the error rate as in the work of Haeupler, and in the client-server setting it also depends on the number of servers. Furthermore, since our approach does not remember the complete transcript history, our current technique can defend only against oblivious corruptions.

The problem of finding densest subgraphs has received increasing attention in recent years finding applications in biology, finance, as well as social network analysis. The k -clique densest subgraph problem is a generalization of the densest subgraph problem, where the objective is to find a subgraph maximizing the ratio between the number of k -cliques in the subgraph and its number of nodes. It includes as a special case the problem of finding subgraphs with largest average number of triangles ( k = 3), which plays an important role in social network analysis. Moreover, algorithms that deal with larger values of k can effectively find quasi-cliques. The densest subgraph problem can be solved in polynomial time with algorithms based on maximum flow, linear programming or a recent approach based on convex optimization. In particular, the latter approach can scale to graphs containing tens of billions of edges. While finding a densest subgraph in large graphs is no longer a bottleneck, the k -clique densest subgraph remains challenging even when k = 3. Our work aims at developing near-optimal and exact algorithms for the k -clique densest subgraph problem on large real-world graphs. We give a surprisingly simple procedure that can be employed to find the maximal k -clique densest subgraph in large-real world graphs. By leveraging appealing properties of existing results, we combine it with a recent approach for listing all k -cliques in a graph and a sampling scheme, obtaining the state-of-the-art approaches for the aforementioned problem. Our theoretical results are complemented with an extensive experimental evaluation showing the effectiveness of our approach in large real-world graphs.

We present a unified (randomized) polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be a tree for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known. Our unified PTAS is based on the previous dynamic programming frameworks proposed in Talwar (STOC 2004) and Bartal, Gottlieb, Krauthgamer (STOC 2012). However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions.

In this article, we design algorithms to maintain approximate core values in dynamic hypergraphs. This notion has been well studied for normal graphs in both static and dynamic setting. We generalize the problem to hypergraphs when edges can be inserted or deleted by an adversary. We consider two dynamic scenarios. In the first case, there are only insertions; and in the second case, there can be both insertions and deletions. In either case, the update time is poly-logarithmic in the number of nodes, with the insertion-only case boasting a better approximation ratio. We also perform extensive experiments on large real-world datasets, which demonstrate the accuracy and efficiency of our algorithms.

We revisit semi-supervised learning on hypergraphs. Same as previous approaches, our method uses a convex program whose objective function is not everywhere differentiable. We exploit the non-uniqueness of the optimal solutions, and consider confidence intervals which give the exact ranges that unlabeled vertices take in any optimal solution. Moreover, we give a much simpler approach for solving the convex program based on the subgradient method. Our experiments on real-world datasets confirm that our confidence interval approach on hypergraphs outperforms existing methods, and our subgradient method gives faster running times when the number of vertices is much larger than the number of edges. Our experiments also support that using directed hypergraphs to capture causal relationships can improve the prediction accuracy. Furthermore, our model can be readily extended to capture multiclass learning.

We design distributed algorithms to compute approximate solutions for several related graph optimization problems. All our algorithms have round complexity being logarithmic in the number of nodes of the underlying graph and in particular independent of the graph diameter. By using a primal-dual approach, we develop a 2(1 + ε)-approximation algorithm for computing the coreness values of the nodes in the underlying graph, as well as a 2(1 + ε)-approximation algorithm for the min-max edge orientation problem, where the goal is to orient the edges so as to minimize the maximum weighted in-degree. We provide lower bounds showing that the aforementioned algorithms are tight both in terms of the approximation guarantee and the round complexity. Finally, motivated by the fact that the densest subset problem has an inherent dependency on the diameter of the graph, we study a weaker version that does not suffer from the same limitation.

As Byzantine Agreement (BA) protocols find application in large-scale decentralized cryptocurrencies, an increasingly important problem is to design BA protocols with improved communication complexity. A few existing works have shown how to achieve subquadratic BA under an adaptive adversary. Intriguingly, they all make a common relaxation about the adaptivity of the attacker, that is, if an honest node sends a message and then gets corrupted in some round, the adversary cannot erase the message that was already sent - henceforth we say that such an adversary cannot perform "after-the-fact removal". By contrast, many (super-)quadratic BA protocols in the literature can tolerate after-the-fact removal. In this paper, we first prove that disallowing after-the-fact removal is necessary for achieving subquadratic-communication BA.

It is well-known that a program's memory access pattern can leak information about its input. To thwart such leakage, most existing works adopt the technique of oblivious RAM (ORAM) simulation. Such an obliviousness notion has stimulated much debate. Although ORAM techniques have significantly improved over the past few years, the concrete overheads are arguably still undesirable for real-world systems — part of this overhead is in fact inherent due to a well-known logarithmic ORAM lower bound by Goldreich and Ostrovsky. To make matters worse, when the program's runtime or output length depend on secret inputs, it may be necessary to perform worst-case padding to achieve full obliviousness and thus incur possibly super-linear overheads. Inspired by the elegant notion of differential privacy, we initiate the study of a new notion of access pattern privacy, which we call “(∊, δ)-differential obliviousness”. We separate the notion of (∊, δ)-differential obliviousness from classical obliviousness by considering several fundamental algorithmic abstractions including sorting small-length keys, merging two sorted lists, and range query data structures (akin to binary search trees). We show that by adopting differential obliviousness with reasonable choices of ∊ and δ, not only can one circumvent several impossibilities pertaining to full obliviousness, one can also, in several cases, obtain meaningful privacy with little overhead relative to the non-private baselines (i.e., having privacy “almost for free”). On the other hand, we show that for very demanding choices of ∊ and δ, the same lower bounds for oblivious algorithms would be preserved for (∊, δ)-differential obliviousness.

We revisit the opinion susceptibility problem that was proposed by Abebe et al. [1], in which agents influence one another's opinions through an iterative process. Each agent has some fixed innate opinion. In each step, the opinion of an agent is updated to some convex combination between its innate opinion and the weighted average of its neighbors' opinions in the previous step. The resistance of an agent measures the importance it places on its innate opinion in the above convex combination. Under non-trivial conditions, this iterative process converges to some equilibrium opinion vector. For the unbudgeted variant of the problem, the goal is to select the resistance of each agent (from some given range) such that the sum of the equilibrium opinions is minimized.

Although external-memory sorting has been a classical algorithms abstraction and has been heavily studied in the literature, perhaps somewhat surprisingly, when data-obliviousness is a requirement, even very rudimentary questions remain open. Prior to our work, it is not even known how to construct a comparison-based, external-memory oblivious sorting algorithm that is optimal in IO-cost. We make a significant step forward in our understanding of external-memory, oblivious sorting algorithms. Not only do we construct a comparison-based, external-memory oblivious sorting algorithm that is optimal in IO-cost, our algorithm is also cache-agnostic in that the algorithm need not know the storage hierarchy's internal parameters such as the cache and cache-line sizes. Our result immediately implies a cache-agnostic ORAM construction whose asymptotic IO-cost matches the best known cache-aware scheme. Last but not the least, we propose and adopt a new and stronger security notion for external-memory, oblivious algorithms and argue that this new notion is desirable for resisting possible cache-timing attacks. Thus our work also lays a foundation for the study of oblivious algorithms in the cache-agnostic model.

Static and dynamic clustering algorithms are a fundamental tool in any machine learning library. Most of the efforts in developing dynamic machine learning and data mining algorithms have been focusing on the sliding window model (where at any given point in time only the most recent data items are retained) or more simplistic models. However, in many real-world applications one might need to deal with arbitrary deletions and insertions. For example, one might need to remove data items that are not necessarily the oldest ones, because they have been flagged as containing inappropriate content or due to privacy concerns. Clustering trajectory data might also require to deal with more general update operations. We develop a (2+ε)-approximation algorithm for the k-center clustering problem with "small»» amortized cost under the fully dynamic adversarial model. In such a model, points can be added or removed arbitrarily, provided that the adversary does not have access to the random choices of our algorithm. The amortized cost of our algorithm is poly-logarithmic when the ratio between the maximum and minimum distance between any two points in input is bounded by a polynomial, while k and epsilon are constant. Our theoretical results are complemented with an extensive experimental evaluation on dynamic data from Twitter, Flickr, as well as trajectory data, demonstrating the effectiveness of our approach.

The celebrated Cheeger’s Inequality (Alon and Milman 1985; Alon 1986) establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. In this article, we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is nonlinear, we have to exploit other properties of the diffusion process to recover the Cheeger’s Inequality that relates hyperedge expansion with the “second eigenvalue” of the resulting Laplacian. However, we show that higher-order spectral properties cannot hold in general using the current framework. Since higher-order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher-order Cheeger-like inequalities. For any k ∈ N, we give a polynomial-time algorithm to compute an O (log r )-approximation to the k th procedural minimizer, where r is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis (introduced by Raghavendra and Steurer (2010)) for constant values of k . Moreover, using the factor-preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs.

We present Path ORAM, an extremely simple Oblivious RAM protocol with a small amount of client storage. Partly due to its simplicity, Path ORAM is the most practical ORAM scheme known to date with small client storage. We formally prove that Path ORAM has a O (log N ) bandwidth cost for blocks of size B = Ω (log 2 N ) bits. For such block sizes, Path ORAM is asymptotically better than the best-known ORAM schemes with small client storage. Due to its practicality, Path ORAM has been adopted in the design of secure processors since its proposal.

Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are essentially oblivious to the input graph. Any greedy algorithm can achieve a performance ratio of 0.5, which is the expected number of matched nodes to the number of nodes in a maximum matching. Since Aronson, Dyer, Frieze, and Suen [ Random Structures Algorithm, 6 (1991), pp. 29--46] proved that the modified randomized greedy algorithm achieves a performance ratio of $0.5 + \epsilon$ (where $\epsilon = \frac{1}{400000}$) on arbitrary graphs in the midnineties, no further attempts in the literature have been made to improve this theoretical ratio for arbitrary graphs until two papers were published in FOCS 2012 [G. Goel and P. Tripathi, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 718--727; M. Poloczek and M. Szegedy, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 708--717]. In this paper, we revisit the ranking algorithm using the linear programming framework. Special care is given to analyze the structural properties of the ranking algorithm in order to derive the linear programming constraints, of which one known as the boundary constraint requires totally new analysis and is crucial to the success of our linear program (LP). We use continuous linear programming relaxation to analyze the limiting behavior as the finite LP grows. Of particular interest are new duality and complementary slackness characterizations that can handle the monotone and the boundary constraints in continuous linear programming. Improving previous work, this paper achieves a theoretical performance ratio of $\frac{2(5-\sqrt{7})}{9} \approx 0.523$ on arbitrary graphs.

We prove the first non-trivial performance ratio strictly above 0.5 for the weighted Ranking algorithm on the oblivious matching problem where nodes in a general graph can have arbitrary weights. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the weighted and the unweighted versions of the problem. Using a new class of continuous linear programming (LP), we prove that the ratio for the weighted case is at least 0.501512, and we improve the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). Unlike previous continuous LP, in which the primal solution must be continuous everywhere, our new continuous LP framework allows the monotone component of the primal function to have jump discontinuities, and the other primal components to take non-conventional forms, such as the Dirac δ function.

We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For k -uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches 1/α ∞ ≈ 0.3178 as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015), and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased. Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k -uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio 1/α ∞ ≈ 0.3178.

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find the shortest tour that visits each of a given collection of subsets (regions or neighborhoods) in the underlying metric space. We give a randomized polynomial-time approximation scheme (PTAS) when the regions are fat weakly disjoint. This notion of regions was first defined when a QPTAS was given for the problem in SODA 2010 (Chan and Elbassioni 2010). The regions are partitioned into a constant number of groups, where in each group, regions should have a common upper bound on their diameters and each region designates one point within it such that these points are far away from one another. We combine the techniques in the previous work, together with the recent PTAS for TSP (STOC 2012: Bartal, Gottlieb, and Krauthgamer 2012) to achieve a PTAS for TSPN. However, several nontrivial technical hurdles need to be overcome for applying the PTAS framework to TSPN: (1) Heuristic to detect sparse instances. In the STOC 2012 paper, a minimum spanning tree heuristic is used to estimate the portion of an optimal tour within some ball. However, for TSPN, it is not known if an optimal tour would use points inside the ball to visit regions that intersect the ball. (2) Partially cut regions in the recursion. After a sparse ball is identified by the heuristic, the PTAS framework for TSP uses dynamic programming to solve the instance restricted to the sparse ball and recurse on the remaining instance. However, for TSPN, it is an important issue to decide whether each region partially intersecting the sparse ball should be solved in the sparse instance or considered in the remaining instance. Surprisingly, we show that both issues can be resolved by conservatively making the ball in question responsible for all intersecting regions. In particular, a sophisticated charging argument is needed to bound the cost of combining tours in the recursion. Moreover, more refined procedures are used to improve the dependence of the running time on the doubling dimension k from the previous exp[( O (1)) k 2 ] (even for just TSP) to exp[2 O ( k log k ) ].

In this paper we study the densest subgraph problem, which plays a key role in many graph mining applications. The goal of the problem is to find a subset of nodes that induces a graph with maximum average degree. The problem has been extensively studied in the past few decades under a variety of different settings. Several exact and approximation algorithms were proposed. However, as normal graph can only model objects with pairwise relationships, the densest subgraph problem fails in identifying communities under relationships that involve more than 2 objects, e.g., in a network connecting authors by publications.

We consider the online vector packing problem in which we have a d dimensional knapsack and items u with weight vectors w_u in R_+^d arrive online in an arbitrary order. Upon the arrival of an item, the algorithm must decide immediately whether to discard or accept the item into the knapsack. When item u is accepted, w_u(i) units of capacity on dimension i will be taken up, for each i in [d]. To satisfy the knapsack constraint, an accepted item can be later disposed of with no cost, but discarded or disposed of items cannot be recovered. The objective is to maximize the utility of the accepted items S at the end of the algorithm, which is given by f(S) for some non-negative monotone submodular function f. For any small constant epsilon > 0, we consider the special case that the weight of an item on every dimension is at most a (1- epsilon) fraction of the total capacity, and give a polynomial-time deterministic O(k / epsilon^2)-competitive algorithm for the problem, where k is the (column) sparsity of the weight vectors. We also show several (almost) tight hardness results even when the algorithm is computationally unbounded. We first show that under the epsilon-slack assumption, no deterministic algorithm can obtain any o(k) competitive ratio, and no randomized algorithm can obtain any o(k / log k) competitive ratio. We then show that for the general case (when epsilon = 0), no randomized algorithm can obtain any o(k) competitive ratio. In contrast to the (1+delta) competitive ratio achieved in Kesselheim et al. [STOC 2014] for the problem with random arrival order of items and under large capacity assumption, we show that in the arbitrary arrival order case, even when |w_u|_infinity is arbitrarily small for all items u, it is impossible to achieve any o(log k / log log k) competitive ratio.

We revisit semi-supervised learning on hypergraphs. Same as previous approaches, our method uses a convex program whose objective function is not everywhere differentiable. We exploit the non-uniqueness of the optimal solutions, and consider confidence intervals which give the exact ranges that unlabeled vertices take in any optimal solution. Moreover, we give a much simpler approach for solving the convex program based on the subgradient method. Our experiments on real-world datasets confirm that our confidence interval approach on hypergraphs outperforms existing methods, and our sub-gradient method gives faster running times when the number of vertices is much larger than the number of edges.

We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For k-uniform matroids, we give a deterministic algorithm with competitive ratio at least 0.2959, and the ratio approaches as k approaches infinity, improving the previous best ratio of 0.25 by Chakrabarti and Kale (IPCO 2014), Buchbinder et al. (SODA 2015) and Chekuri et al. (ICALP 2015). We also show that our algorithm is optimal among a class of deterministic monotone algorithms that accept a new arriving element only if the objective is strictly increased. Further, we prove that no deterministic monotone algorithm can be strictly better than 0.25-competitive even for partition matroids, the most modest generalization of k-uniform matroids, matching the competitive ratio by Chakrabarti and Kale (IPCO 2014) and Chekuri et al. (ICALP 2015). Interestingly, we show that randomized algorithms are strictly more powerful by giving a (non-monotone) randomized algorithm for partition matroids with ratio Finally, our techniques can be extended to a more general problem that generalizes both the online sub- modular maximization problem and the online bipartite matching problem with free disposal. Using the techniques developed in this paper, we give constant- competitive algorithms for the submodular online bipartite matching problem.

Algorithms for finding dense regions in an input graph have proved to be effective tools in graph mining and data analysis. Recently, Tatti and Gionis [WWW 2015] presented a novel graph decomposition (known as the locally-dense decomposition) that is similar to the well-known k-core decomposition, with the additional property that its components are arranged in order of their densities. Such a decomposition provides a valuable tool in graph mining. Unfortunately, their algorithm for computing the exact decomposition is based on a maximum-flow algorithm which cannot scale to massive graphs, while the approximate decomposition defined by the same authors misses several interesting properties. This calls for scalable algorithms for computing such a decomposition. In our work, we devise an efficient algorithm which is able to compute exact locally-dense decompositions in real-world graphs containing up to billions of edges. Moreover, we provide a new definition of approximate locally-dense decomposition which retains most of the properties of an exact decomposition, for which we devise an algorithm that can scale to real-world graphs containing up to tens of billions of edges. Our algorithm is based on the classic Frank-Wolfe algorithm which is similar to gradient descent and can be efficiently implemented in most of the modern architectures dealing with massive graphs. We provide a rigorous study of our algorithms and their convergence rates. We conduct an extensive experimental evaluation on multi-core architectures showing that our algorithms converge much faster in practice than their worst-case analysis. Our algorithm is even more efficient for the more specialized problem of computing a densest subgraph.

We consider a distributed private data analysis setting, where multiple parties each hold some sensitive data and they wish to run a protocol to learn some aggregate statistics over the distributed dataset, while protecting each user’s privacy. As an initial effort, we consider a distributed summation problem. We first show a lower bound, that is, under information-theoretic differential privacy, any multi-party protocol with a small number of messages must have large additive error. We then show that by adopting a computational differential privacy notion, one can circumvent this lower bound and design practical protocols for the periodic distributed summation problem. Our construction has several desirable features. First, it works in the client-server model and requires no peer-to-peer communication among the clients. Second, our protocol is fault tolerant and can output meaningful statistics even when a subset of the participants fail to respond. Our constructions guarantee the privacy of honest parties even when a fraction of the participants may be compromised and colluding. In addition, we propose a new distributed noise addition mechanism that guarantees small total error.

We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem. Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs). (1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). (2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as \emph{matching coverage}, in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied. Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5.

We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: minxϵR+nf(x) s.t. Ax ≥ 1, where f:R+n→ R+is a monotone convex function, and A is an m×n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We also consider a convex packing problem defined as: maxyϵR+mΣj=1myj - g(ATy), where g:R+n→R+is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f. We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.

We achieve a (randomized) polynomial-time approximation scheme (PTAS) for the Steiner forest problem in doubling metrics. Before our work, a PTAS was given only for the Euclidean plane in [G. Borradaile, P. N. Klein, and C. Mathieu, in FOCS, IEEE Computer Society, 2008, pp. 115--124]. Our PTAS also shares similarities with the dynamic programming for sparse instances used in [Y. Bartal, L. Gottlieb, and R. Krauthgamer, in STOC, ACM, 2012, pp. 663--672] and [T-H. H. Chan and S.-H. Jiang, in SODA, SIAM, 2016, pp. 754--765]. However, extending previous approaches requires overcoming several nontrivial hurdles, and we make the following technical contributions. (1) We prove a technical lemma showing that Steiner points have to be “near” the terminals in an optimal Steiner tree. This enables us to define a heuristic to estimate the local behavior of the optimal solution, even though the Steiner points are unknown in advance. This lemma also generalizes previous results in the Euclidean plane and may be of independent interest for related problems involving Steiner points. (2) We develop a novel algorithmic technique known as “adaptive cells” to overcome the difficulty of keeping track of multiple components in a solution. Our idea is based on but significantly different from the previously proposed “uniform cells” in [G. Borradaile, P. N. Klein, and C. Mathieu, in FOCS, IEEE Computer Society, 2008, pp. 115--124], where techniques cannot be readily applied to doubling metrics.

We study the max-min fair allocation problem in which a set of m indivisible items are to be distributed among n agents such that the minimum utility among all agents is maximized. In the restricted setting, the utility of each item j on agent i is either 0 or some non-negative weight w_j. For this setting, Asadpour et al. [TALG, 2012] showed that a certain configuration-LP can be used to estimate the optimal value within a factor of 4 + delta, for any delta > 0, which was recently extended by Annamalai et al. [SODA 2015] to give a polynomial-time 13-approximation algorithm for the problem. For hardness results, Bezáková and Dani [SIGecom Exch., 2005] showed that it is NP-hard to approximate the problem within any ratio smaller than 2. In this paper we consider the (1, epsilon)-restricted max-min fair allocation problem, in which for some parameter epsilon in (0, 1), each item j is either heavy (w_j = 1) or light (w_j = epsilon). We show that the (1, epsilon)-restricted case is also NP-hard to approximate within any ratio smaller than 2. Hence, this simple special case is still algorithmically interesting. Using the configuration-LP, we are able to estimate the optimal value of the problem within a factor of 3 + delta, for any delta > 0. Extending this idea, we also obtain a quasi-polynomial time (3 + 4 epsilon)-approximation algorithm and a polynomial time 9-approximation algorithm. Moreover, we show that as epsilon tends to 0, the approximation ratio of our polynomial-time algorithm approaches 3 + 2 sqrt{2} approx 5.83.

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a given collection of subsets (regions or neighborhoods) in the underlying metric space. We give a randomized polynomial time approximation scheme (PTAS) when the regions are fat weakly disjoint. This notion of regions was first defined when a QPTAS was given for the problem in [SODA 2010: Chan and Elbassioni]. We combine the techniques in the previous work, together with the recent PTAS for TSP [STOC 2012: Bartal, Gottlieb and Krauthgamer] to achieve a PTAS for TSPN. Moreover, more refined procedures are used to improve the dependence of the running time on the doubling dimension k from the previous exp[O(1)k2] (even for just TSP) to exp[O(1)O(k log k)].

We study the problem of routing in doubling metrics and show how to perform hierarchical routing in such metrics with small stretch and compact routing tables (i.e., with a small amount of routing information stored at each vertex). We say that a metric ( X , d ) has doubling dimension dim(α balls of half its radius. (A doubling metric is one whose doubling dimension dim(G . We show how to perform (1 + τ)-stretch routing on such a metric for any 0 < τ ≤ 1 with routing tables of size at most (α/τ) O (α) log Δlog δ bits with only (α/τ) O (α) log Δ entries , where Δ is the diameter of the graph, and δ is the maximum degree of the graph G ; hence, the number of routing table entries is just τ − O (1) log Δ for doubling metrics. These results extend and improve on those of Talwar (2004). We also give better constructions of sparse spanners for doubling metrics than those obtained from the routing tables earlier; for τ > 0, we give algorithms to construct (1 + τ)-stretch spanners for a metric ( X , d ) with maximum degree at most (2 + 1/τ) O(dim(X)) , matching the results of Das et al. for Euclidean metrics.

We propose a new tree-based ORAM scheme called Circuit ORAM. Circuit ORAM makes both theoretical and practical contributions. From a theoretical perspective, Circuit ORAM shows that the well-known Goldreich-Ostrovsky logarithmic ORAM lower bound is tight under certain parameter ranges, for several performance metrics. Therefore, we are the first to give an answer to a theoretical challenge that remained open for the past twenty-seven years. Second, Circuit ORAM earns its name because it achieves (almost) optimal circuit size both in theory and in practice for realistic choices of block sizes. We demonstrate compelling practical performance and show that Circuit ORAM is an ideal candidate for secure multi-party computation applications.

We consider the general (J, K)-secretary problem, where n totally ordered items arrive in a random order. An algorithm observes the relative merits of arriving items and is allowed to make J selections. The objective is to maximize the expected number of items selected among the K best items. Buchbinder, Jain and Singh proposed a finite linear program (LP) that completely characterizes the problem, but it is difficult to analyze the asymptotic behavior of its optimal solution as n tends to infinity. Instead, we prove a formal connection between the finite model and an infinite model, where there are a countably infinite number of items, each of which has arrival time drawn independently and uniformly from [0, 1]. The finite LP extends to a continuous LP, whose complementary slackness conditions reveal an optimal algorithm which involves JK thresholds that play a similar role as the -threshold in the optimal classical secretary algorithm. In particular, for the case K = 1, the J optimal thresholds have a nice “rational description”. Our continuous LP analysis gives a very clear perspective on the problem, and the new insights inspire us to solve two related problems. 1. We settle the open problem whether algorithms based only on relative merits can achieve optimal ratio for matroid secretary problems. We show that, for online 2-item auction with random arriving bids (the K-uniform matroid problem with K = 2), an algorithm making decisions based only on relative merits cannot achieve the optimal ratio. This is in contrast with the folklore that, for online 1-item auction, no algorithm can have performance ratio strictly larger than 1/e, which is achievable by an algorithm that considers only relative merits. 2. We give a general transformation technique that takes any monotone algorithm (such as threshold algorithms) for the (K, K)-secretary problem, and constructs an algorithm for online bipartite K-matching with random arrival order that has at least the same performance guarantee.

Finding dense subgraphs in large graphs is a key primitive in a variety of real-world application domains, encompassing social network analytics, event detection, biology, and finance. In most such applications, one typically aims at finding several (possibly overlapping) dense subgraphs which might correspond to communities in social networks or interesting events. While a large amount of work is devoted to finding a single densest subgraph, perhaps surprisingly, the problem of finding several dense subgraphs with limited overlap has not been studied in a principled way, to the best of our knowledge. In this work we define and study a natural generalization of the densest subgraph problem, where the main goal is to find at most $k$ subgraphs with maximum total aggregate density, while satisfying an upper bound on the pairwise Jaccard coefficient between the sets of nodes of the subgraphs. After showing that such a problem is NP-Hard, we devise an efficient algorithm that comes with provable guarantees in some cases of interest, as well as, an efficient practical heuristic. Our extensive evaluation on large real-world graphs confirms the efficiency and effectiveness of our algorithms.

In a seminal STOC 1995 paper, Arya et al. conjectured that spanners for low-dimensional Euclidean spaces with constant maximum degree, hop-diameter $O(\log n)$, and lightness $O(\log n)$ (i.e., weight $O(\log n) \cdot w({MST}))$ can be constructed in $O(n \log n)$ time. This conjecture, which became a central open question in this area, was resolved in the affirmative by Elkin and Solomon in STOC 2013. In fact, Elkin and Solomon proved that the conjecture of Arya et al. holds even in doubling metrics. However, Elkin and Solomon's spanner construction is complicated. In this work we present a significantly simpler construction of spanners for doubling metrics with the same guarantees as above. Our construction is based on the basic net-tree spanner framework. However, by employing well-known properties of the net-tree spanner in conjunction with numerous new ideas, we managed to get significantly stronger results. First and foremost, our construction extends in a simple and natural way to provide $k$-fault tolerant spanners with maximum degree $O(k^2)$, hop-diameter $O(\log n)$, and lightness $O(k^2 \log n)$. This is the first construction of fault-tolerant spanners (even for Euclidean metrics) that achieves good bounds (polylogarithmic in $n$ and polynomial in $k$) on all the involved parameters simultaneously. Second, we show that the lightness bound of our construction can be improved to $O(k^2)$ (with high probability), for random points in $[0,1]^D$, where $2 \le D = O(1)$.

Oblivious RAMs (ORAMs) have traditionally been measured by their bandwidth overhead and client storage. We observe that when using ORAMs to build secure computation protocols for RAM programs, the size of the ORAM circuits is more relevant to the performance.

We design novel, asymptotically more efficient data structures and algorithms for programs whose data access patterns exhibit some degree of predictability. To this end, we propose two novel techniques, a pointer-based technique and a locality-based technique. We show that these two techniques are powerful building blocks in making data structures and algorithms oblivious. Specifically, we apply these techniques to a broad range of commonly used data structures, including maps, sets, priority-queues, stacks, deques; and algorithms, including a memory allocator algorithm, max-flow on graphs with low doubling dimension, and shortest-path distance queries on weighted planar graphs. Our oblivious counterparts of the above outperform the best known ORAM scheme both asymptotically and in practice.

P2P file streaming has become very popular in online video sharing. Under the video on demand (VoD) setting, peers in the network may be interested in different portions of the same video. It is a new challenge to distribute the server load to peers under the VoD setting while at the same time maintaining the streaming performance, i.e., low latency and good fluency. Our approach is to use techniques in social recommendation to spread information on which portions are currently popular. However, peers might not always reveal the truth, because of privacy issues or selfish behavior. In this paper, we describe and discuss a synchronized large-scale P2P VoD system in which each peer can only communicate with one other peer in one round. We assume the video to be streamed is divided into M consecutive chunks and only one chunk can be transmitted due to network bandwidth every communication. We decentralize the whole system such that each peer has no extra information about the network or how other peers behave, and can only communicate with its own neighbors independently without the help of tracker to obtain robustness. Moreover, the model we consider is fully dynamic: peers leave and join the network frequently. We show by experiment that even under the bounded connection, bounded transmission and distributed setting, our protocol ensures that almost all peers in the dynamic P2P VoD network can achieve low latency and good fluency in different kinds of network topologies. We also analyze the streaming performance when peers behave differently (selfish vs. unselfish). We show that peers have little incentive to be selfish in our protocol, which means that our protocol is in a sense self-enforcing.

Motivated by online advertisement and exchange settings, greedy randomized algorithms for the maximum matching problem have been studied, in which the algorithm makes (random) decisions that are essentially oblivious to the input graph. Any greedy algorithm can achieve performance ratio 0.5, which is the expected number of matched nodes to the number of nodes in a maximum matching. Since Aronson, Dyer, Frieze and Suen proved that the Modified Randomized Greedy algorithm achieves performance ratio 0.5+ ∊ (where ) on arbitrary graphs in the mid-nineties, no further attempts in the literature have been made to improve this theoretical ratio for arbitrary graphs until two papers were published in FOCS 2012. In this paper, we revisit the Ranking algorithm using the LP framework. Special care is given to analyze the structural properties of the Ranking algorithm in order to derive the LP constraints, of which one known as the boundary constraint requires totally new analysis and is crucial to the success of our LP. We use continuous LP relaxation to analyze the limiting behavior as the finite LP grows. Of particular interest are new duality and complementary slackness characterizations that can handle the monotone and the boundary constraints in continuous LP. Our work achieves the currently best theoretical performance ratio of on arbitrary graphs. Moreover, experiments suggest that Ranking cannot perform better than 0.724 in general.

In this article, we study the convergence time required to achieve consensus in dynamic networks. In each timestep, a node's value is updated to some weighted average of its neighbors and its old values. We study the case when the underlying network is dynamic and investigate different averaging models. Both our analysis and experiments show that dynamic networks exhibit fast convergence behavior, even under very mild connectivity assumptions.

The traveling salesman problem (TSP) is a canonical NP-complete problem which is proved by Trevisan [SIAM J. Comput., 30 (2000), pp. 475--485] to be MAX-SNP hard even on high-dimensional Euclidean metrics. To circumvent this hardness, researchers have been developing approximation schemes for „simpler” instances of the problem. For instance, the algorithms of Arora and of Talwar show how to approximate TSP on low-dimensional metrics (for different notions of metric dimension). However, a feature of most current notions of metric dimension is that they are „local”: the definitions require every local neighborhood to be well-behaved. In this paper, we define a global notion of dimension that generalizes the popular notion of doubling dimension, but still allows some small dense regions; e.g., it allows some metrics that contain cliques of size $\sqrt{n}$. Given a metric with global dimension $\dim_{C}$, we give a $(1+\varepsilon)$-approximation algorithm that runs in subexponential time, i.e., in $\exp(O(n^{\delta}\varepsilon^{-4\dim_{C}}))$-time for every constant $0<\delta<1$. As mentioned above, metrics with bounded $\dim_{C}$ may contain metrics of size $O(\sqrt{n})$ on which the TSP problem is hard to approximate to within $(1+\varepsilon)$. Hence, to do better than a running time of $\Omega(\exp\{\sqrt{n}\})$, our algorithms find $O(1)$-approximations to some portions of the tour, and $(1+\varepsilon)$-approximations for other portions, and stitch them together. Moreover, we show that such globally bounded metrics have spanners that preserve distances to arbitrary accuracy and have size $\Theta(n^{1.5})$.

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of n subsets ( regions or neighborhoods ) in the underlying metric space. We give a quasi-polynomial time approximation scheme (QPTAS) when the regions are what we call α - fat weakly disjoint . This notion combines the existing notions of diameter variation, fatness and disjointness for geometric objects and generalizes these notions to any arbitrary metric space. Intuitively, the regions can be grouped into a bounded number of types, where in each type, the regions have similar upper bounds for their diameters, and each such region can designate a point such that these points are far away from one another. Our result generalizes the polynomial time approximation scheme (PTAS) for TSPN on the Euclidean plane by Mitchell (in SODA, pp. 11–18, 2007 ) and the QPTAS for TSP on doubling metrics by Talwar (in 36th STOC, pp. 281–290, 2004 ). We also observe that our techniques directly extend to a QPTAS for the Group Steiner Tree Problem on doubling metrics, with the same assumption on the groups.

We ask the question: how can Web sites and data aggregators continually release updated statistics, and meanwhile preserve each individual user’s privacy? Suppose we are given a stream of 0’s and 1’s. We propose a differentially private continual counter that outputs at every time step the approximate number of 1’s seen thus far. Our counter construction has error that is only poly-log in the number of time steps. We can extend the basic counter construction to allow Web sites to continually give top- k and hot items suggestions while preserving users’ privacy.

Sources of data uncertainty and imprecision are numerous. A way to handle this uncertainty is to associate probabilistic annotations to data. Many such probabilistic database models have been proposed, both in the relational and in the semi-structured setting. The latter is particularly well adapted to the management of uncertain data coming from a variety of automatic processes. An important problem, in the context of probabilistic XML databases, is that of answering aggregate queries (count, sum, avg, etc.), which has received limited attention so far. In a model unifying the various (discrete) semi-structured probabilistic models studied up to now, we present algorithms to compute the distribution of the aggregation values (exploiting some regularity properties of the aggregate functions) and probabilistic moments (especially expectation and variance) of this distribution. We also prove the intractability of some of these problems and investigate approximation techniques. We finally extend the discrete model to a continuous one, in order to take into account continuous data values, such as measurements from sensor networks, and extend our algorithms and complexity results to the continuous case.

Sources of data uncertainty and imprecision are numerous. A way to handle this uncertainty is to associate probabilistic annotations to data. Many such probabilistic database models have been proposed, both in the relational and in the semi-structured setting. The latter is particularly well adapted to the management of uncertain data coming from a variety of automatic processes. An important problem, in the context of probabilistic XML databases, is that of answering aggregate queries (count, sum, avg, etc.), which has received limited attention so far. In a model unifying the various (discrete) semi-structured probabilistic models studied up to now, we present algorithms to compute the distribution of the aggregation values (exploiting some regularity properties of the aggregate functions) and probabilistic moments (especially, expectation and variance) of this distribution. We also prove the intractability of some of these problems and investigate approximation techniques. We finally extend the discrete model to a continuous one, in order to take into account continuous data values, such as measurements from sensor networks, and present algorithms to compute distribution functions and moments for various classes of continuous distributions of data values.

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of n subsets (regions or neighborhoods) in the underlying metric space. We give a QPTAS when the regions are what we call α-fat weakly disjoint. This notion combines the existing notions of diameter variation, fatness and disjointness for geometric objects and generalizes these notions to any arbitrary metric space. Intuitively the regions can be grouped into a bounded number of types, where in each type, the regions have similar upper bounds for their diameters, and each such region can designate a point such that these points are far away from one another. Our result generalizes the PTAS for TSPN on the Euclidean plane by Mitchell [27] and the QPTAS for TSP on doubling metrics by Talwar [30]. We also observe that our techniques directly extend to a QPTAS for the Group Steiner Tree Problem on doubling metrics, with the same assumption on the groups.

We consider the problem of embedding a metric into low-dimensional Euclidean space. The classical theorems of Bourgain, and of Johnson and Lindenstrauss say that any metric on n points embeds into an O (log n )-dimensional Euclidean space with O (log n ) distortion. Moreover, a simple “volume” argument shows that this bound is nearly tight: a uniform metric on n points requires nearly logarithmic number of dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to low-dimensional embeddings. In other words, do doubling metrics, that do not have large uniform submetrics, and thus no volume hurdles to low dimensional embeddings, embed in low dimensional Euclidean spaces with small distortion? In this article, we give a positive answer to this question. We show how to embed any doubling metrics into O (log log n ) dimensions with O (log n ) distortion. This is the first embedding for doubling metrics into fewer than logarithmic number of dimensions, even allowing for logarithmic distortion. This result is one extreme point of our general trade-off between distortion and dimension: given an n -point metric (V,d) with doubling dimension dim D , and any target dimension T in the range Ω(dim D log log n ) ≤ T ≤ O (log n ), we show that the metric embeds into Euclidean space R T with O (log n √ dim D / T ) distortion.

The Lovaacutesz thetav-function [Lov79] on a graph G = (V,E) can be defined as the maximum of the sum of the entries of a positive semidefinite matrix X, whose trace Tr(X) equals 1, and Xij= 0 whenever {i, j} isin E. This function appears as a subroutine for many algorithms for graph problems such as maximum independent set and maximum clique. We apply Arora and Kale's primal-dual method for SDP to design an approximate algorithm for the thetav-function with an additive error of delta > 0, which runs in time O(alpha2n2/delta2log n middot Me), where alpha = thetav(G) and Me= O(n3) is the time for a matrix exponentiation operation. Moreover, our techniques generalize to the weighted Lovasz thetav-function, and both the maximum independent set weight and the maximum clique weight for vertex weighted perfect graphs can be approximated within a factor of (1+epsi) in time O(epsi-2n5log n).

Given a metric M=(V,d), a graph G=(V,E) is a t-spanner for M if every pair of nodes in V has a “short” path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every pair of nodes has such a short path that also uses at most D edges. We consider the problem of constructing sparse (1+ε)-spanners with small hop diameter for metrics of low doubling dimension. In this paper, we show that given any metric with constant doubling dimension k and any 0<ε<1, one can find (1+ε)-spanner for the metric with nearly linear number of edges (i.e., only O(nlog * n+n ε −O(k)) edges) and constant hop diameter; we can also obtain a (1+ε)-spanner with linear number of edges (i.e., only n ε −O(k) edges) that achieves a hop diameter that grows like the functional inverse of Ackermann’s function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal.

We consider the problem of embedding finite metrics with slack: We seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler [in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, 2004], we show that provable guarantees of this type can in fact be achieved in general: Any finite metric space can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into $\ell_1$ which exhibit gracefully degrading distortion: There is a single embedding into $\ell_1$ that achieves distortion at most $O(\log\frac{1}{\epsilon})$ on all but at most-1.5pt an $\epsilon$ fraction of distances simultaneously for all $\epsilon>0$. We extend this with distortion1pt $O(\log\frac{1}{\epsilon})^{1/p}$ to maps into general $\ell_p$, $p\geq1$, for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight and give a general technique to obtain lower bounds for $\epsilon$-slack embeddings from lower bounds for low-distortion embeddings.

We design an encryption scheme called Multi-dimensional Range Query over Encrypted Data (MRQED), to address the privacy concerns related to the sharing of network audit logs and various other applications. Our scheme allows a network gateway to encrypt summaries of network flows before submitting them to an untrusted repository. When network intrusions are suspected, an authority can release a key to an auditor, allowing the auditor to decrypt flows whose attributes (e.g., source and destination addresses, port numbers, etc.) fall within specific ranges. However, the privacy of all irrelevant flows are still preserved. We formally define the security for MRQED and prove the security of our construction under the decision bilinear Diffie-Hellman and decision linear assumptions in certain bilinear groups. We study the practical performance of our construction in the context of network audit logs. Apart from network audit logs, our scheme also has interesting applications for financial audit logs, medical privacy, untrusted remote storage, etc. In particular, we show that MRQED implies a solution to its dual problem, which enables investors to trade stocks through a broker in a privacypreserving manner.

We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler (2004), we show that provable guarantees of this type can in fact be achieved in general: any finite metric can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into /spl lscr//sub 1/ which exhibit gracefully degrading distortion: these is a single embedding into /spl lscr//sub 1/ that achieves distortion at most O(log 1//spl epsi/) on all but at most an /spl epsi/ fraction of distances, simultaneously for all /spl epsi/ > 0. We extend this with distortion O(log 1//spl epsi/)/sup 1/p/ to maps into general /spl lscr//sub p/, p /spl ges/ 1 for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight, and give a general technique to obtain lower bounds for /spl epsi/-slack embeddings from lower bounds for low-distortion embeddings.