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We consider the question of orienting the edges in a graph $G$ such that every vertex has bounded out-degree. For graphs of arboricity $\alpha$, there is an orientation in which every vertex has out-degree at most $\alpha$ and, moreover, the best possible maximum out-degree of an orientation is at least $\alpha - 1$. We are thus interested in algorithms that can achieve a maximum out-degree of close to $\alpha$. A widely studied approach for this problem in the distributed algorithms setting is a ``peeling algorithm'' that provides an orientation with maximum out-degree $\alpha(2+\epsilon)$ in a logarithmic number of iterations. We consider this problem in the local computation algorithm (LCA) model, which quickly answers queries of the form ``What is the orientation of edge $(u,v)$?'' by probing the input graph. When the peeling algorithm is executed in the LCA setting by applying standard techniques, e.g., the Parnas-Ron paradigm, it requires $\Omega(n)$ probes per query on an $n$-vertex graph. In the case where $G$ has unbounded degree, we show that any LCA that orients its edges to yield maximum out-degree $r$ must use $\Omega(\sqrt n/r)$ probes to $G$ per query in the worst case, even if $G$ is known to be a forest (that is, $\alpha=1$). We also show several algorithms with sublinear probe complexity when $G$ has unbounded degree. When $G$ is a tree such that the maximum degree $\Delta$ of $G$ is bounded, we demonstrate an algorithm that uses $\Delta n^{1-\log_\Delta r + o(1)}$ probes to $G$ per query. To obtain this result, we develop an edge-coloring approach that ultimately yields a graph-shattering-like result. We also use this shattering-like approach to demonstrate an LCA which $4$-colors any tree using sublinear probes per query.

We design a deterministic algorithm for the $(1+\epsilon)$-approximate maximum matching problem. Our primary result demonstrates that this problem can be solved in $O(\epsilon^{-6})$ semi-streaming passes, improving upon the $O(\epsilon^{-19})$ pass-complexity algorithm by [Fischer, Mitrovi\'c, and Uitto, STOC'22]. This contributes substantially toward resolving Open question 2 from [Assadi, SOSA'24]. Leveraging the framework introduced in [FMU'22], our algorithm achieves an analogous round complexity speed-up for computing a $(1+\epsilon)$-approximate maximum matching in both the Massively Parallel Computation (MPC) and CONGEST models. The data structures maintained by our algorithm are formulated using blossom notation and represented through alternating trees. This approach enables a simplified correctness analysis by treating specific components as if operating on bipartite graphs, effectively circumventing certain technical intricacies present in prior work.

We consider the problem of counting the copies of a length-$k$ pattern $\sigma$ in a sequence $f \colon [n] \to \mathbb{R}$, where a copy is a subset of indices $i_1 < \ldots < i_k \in [n]$ such that $f(i_j) < f(i_\ell)$ if and only if $\sigma(j) < \sigma(\ell)$. This problem is motivated by a range of connections and applications in ranking, nonparametric statistics, combinatorics, and fine-grained complexity, especially when $k$ is a small fixed constant. Recent advances have significantly improved our understanding of counting and detecting patterns. Guillemot and Marx [2014] demonstrated that the detection variant is solvable in $O(n)$ time for any fixed $k$. Their proof has laid the foundations for the discovery of the twin-width, a concept that has notably advanced parameterized complexity in recent years. Counting, in contrast, is harder: it has a conditional lower bound of $n^{\Omega(k / \log k)}$ [Berendsohn, Kozma, and Marx 2019] and is expected to be polynomially harder than detection as early as $k = 4$, given its equivalence to counting $4$-cycles in graphs [Dudek and Gawrychowski, 2020]. In this work, we design a deterministic near-linear time $(1+\varepsilon)$-approximation algorithm for counting $\sigma$-copies in $f$ for all $k \leq 5$. Combined with the conditional lower bound for $k=4$, this establishes the first known separation between approximate and exact algorithms for pattern counting. Interestingly, our algorithm leverages the Birg\'e decomposition -- a sublinear tool for monotone distributions widely used in distribution testing -- which, to our knowledge, has not been applied in a pattern counting context before.

The SetCover problem has been extensively studied in many different models of computation, including parallel and distributed settings. From an approximation point of view, there are two standard guarantees: an $O(\log \Delta)$-approximation (where $\Delta$ is the maximum set size) and an $O(f)$-approximation (where $f$ is the maximum number of sets containing any given element). In this paper, we introduce a new, surprisingly simple, model-independent approach to solving SetCover in unweighted graphs. We obtain multiple improved algorithms in the MPC and CRCW PRAM models. First, in the MPC model with sublinear space per machine, our algorithms can compute an $O(f)$ approximation to SetCover in $\hat{O}(\sqrt{\log \Delta} + \log f)$ rounds, where we use the $\hat{O}(x)$ notation to suppress $\mathrm{poly} \log x$ and $\mathrm{poly} \log \log n$ terms, and a $O(\log \Delta)$ approximation in $O(\log^{3/2} n)$ rounds. Moreover, in the PRAM model, we give a $O(f)$ approximate algorithm using linear work and $O(\log n)$ depth. All these bounds improve the existing round complexity/depth bounds by a $\log^{\Omega(1)} n$ factor. Moreover, our approach leads to many other new algorithms, including improved algorithms for the HypergraphMatching problem in the MPC model, as well as simpler SetCover algorithms that match the existing bounds.

Given an undirected, weighted $n$-vertex graph $G = (V, E, w)$, a Gomory-Hu tree $T$ is a weighted tree on $V$ such that for any pair of distinct vertices $s, t \in V$, the Min-$s$-$t$-Cut on $T$ is also a Min-$s$-$t$-Cut on $G$. Computing a Gomory-Hu tree is a well-studied problem in graph algorithms and has received considerable attention. In particular, a long line of work recently culminated in constructing a Gomory-Hu tree in almost linear time [Abboud, Li, Panigrahi and Saranurak, FOCS 2023]. We design a differentially private (DP) algorithm that computes an approximate Gomory-Hu tree. Our algorithm is $\varepsilon$-DP, runs in polynomial time, and can be used to compute $s$-$t$ cuts that are $\tilde{O}(n/\varepsilon)$-additive approximations of the Min-$s$-$t$-Cuts in $G$ for all distinct $s, t \in V$ with high probability. Our error bound is essentially optimal, as [Dalirrooyfard, Mitrovi\'c and Nevmyvaka, NeurIPS 2023] showed that privately outputting a single Min-$s$-$t$-Cut requires $\Omega(n)$ additive error even with $(1, 0.1)$-DP and allowing for a multiplicative error term. Prior to our work, the best additive error bounds for approximate all-pairs Min-$s$-$t$-Cuts were $O(n^{3/2}/\varepsilon)$ for $\varepsilon$-DP [Gupta, Roth and Ullman, TCC 2012] and $O(\sqrt{mn} \cdot \text{polylog}(n/\delta) / \varepsilon)$ for $(\varepsilon, \delta)$-DP [Liu, Upadhyay and Zou, SODA 2024], both of which are implied by differential private algorithms that preserve all cuts in the graph. An important technical ingredient of our main result is an $\varepsilon$-DP algorithm for computing minimum Isolating Cuts with $\tilde{O}(n / \varepsilon)$ additive error, which may be of independent interest.

We consider the PageRank problem in the dynamic setting, where the goal is to explicitly maintain an approximate PageRank vector $\pi \in \mathbb{R}^n$ for a graph under a sequence of edge insertions and deletions. Our main result is a complete characterization of the complexity of dynamic PageRank maintenance for both multiplicative and additive ($L_1$) approximations. First, we establish matching lower and upper bounds for maintaining additive approximate PageRank in both incremental and decremental settings. In particular, we demonstrate that in the worst-case $(1/\alpha)^{\Theta(\log \log n)}$ update time is necessary and sufficient for this problem, where $\alpha$ is the desired additive approximation. On the other hand, we demonstrate that the commonly employed ForwardPush approach performs substantially worse than this optimal runtime. Specifically, we show that ForwardPush requires $\Omega(n^{1-\delta})$ time per update on average, for any $\delta > 0$, even in the incremental setting. For multiplicative approximations, however, we demonstrate that the situation is significantly more challenging. Specifically, we prove that any algorithm that explicitly maintains a constant factor multiplicative approximation of the PageRank vector of a directed graph must have amortized update time $\Omega(n^{1-\delta})$, for any $\delta > 0$, even in the incremental setting, thereby resolving a 13-year old open question of Bahmani et al.~(VLDB 2010). This sharply contrasts with the undirected setting, where we show that $\rm{poly}\ \log n$ update time is feasible, even in the fully dynamic setting under oblivious adversary.

Given a graph with positive and negative edge labels, the correlation clustering problem aims to cluster the nodes so to minimize the total number of between-cluster positive and within-cluster negative edges. This problem has many applications in data mining, particularly in unsupervised learning. Inspired by the prevalence of large graphs and constantly changing data in modern applications, we study correlation clustering in dynamic, parallel (MPC), and local computation (LCA) settings. We design an approach that improves state-of-the-art runtime complexities in all these settings. In particular, we provide the first fully dynamic algorithm that runs in an expected amortized constant time, without any dependence on the graph size. Moreover, our algorithm essentially matches the approximation guarantee of the celebrated Pivot algorithm.

Finding min $s$-$t$ cuts in graphs is a basic algorithmic tool with applications in image segmentation, community detection, reinforcement learning, and data clustering. In this problem, we are given two nodes as terminals, and the goal is to remove the smallest number of edges from the graph so that these two terminals are disconnected. We study the complexity of differential privacy for the min $s$-$t$ cut problem and show nearly tight lower and upper bounds where we achieve privacy at no cost for running time efficiency. We also develop a differentially private algorithm for the multiway $k$-cut problem, in which we are given $k$ nodes as terminals that we would like to disconnect. As a function of $k$, we obtain privacy guarantees that are exponentially more efficient than applying the advanced composition theorem to known algorithms for multiway $k$-cut. Finally, we empirically evaluate the approximation of our differentially private min $s$-$t$ cut algorithm and show that it almost matches the quality of the output of non-private ones.

Finding dense subgraphs is a fundamental algorithmic tool in data mining, community detection, and clustering. In this problem, one aims to find an induced subgraph whose edge-to-vertex ratio is maximized. We study the directed case of this question in the context of semi-streaming and massively parallel algorithms. In particular, we show that it is possible to find a $(2+\epsilon)$ approximation on randomized streams even in a single pass by using $O(n \cdot {\rm poly} \log n)$ memory on $n$-vertex graphs. Our result improves over prior works, which were designed for arbitrary-ordered streams: the algorithm by Bahmani et al. (VLDB 2012) which uses $O(\log n)$ passes, and the work by Esfandiari et al. (2015) which makes one pass but uses $O(n^{3/2})$ memory. Moreover, our techniques extend to the Massively Parallel Computation model yielding $O(1)$ rounds in the super-linear and $O(\sqrt{\log n})$ rounds in the nearly-linear memory regime. This constitutes a quadratic improvement over state-of-the-art bounds by Bahmani et al. (VLDB 2012 and WAW 2014), which require $O(\log n)$ rounds even in the super-linear memory regime. Finally, we empirically evaluate our single-pass semi-streaming algorithm on $6$ benchmarks and show that, even on non-randomly ordered streams, the quality of its output is essentially the same as that of Bahmani et al. (VLDB 2012) while it is $2$ times faster on large graphs.

This paper presents an $O(\log\log \bar{d})$ round massively parallel algorithm for $1+\epsilon$ approximation of maximum weighted $b$-matchings, using near-linear memory per machine. Here $\bar{d}$ denotes the average degree in the graph and $\epsilon$ is an arbitrarily small positive constant. Recall that $b$-matching is the natural and well-studied generalization of the matching problem where different vertices are allowed to have multiple (and differing number of) incident edges in the matching. Concretely, each vertex $v$ is given a positive integer budget $b_v$ and it can have up to $b_v$ incident edges in the matching. Previously, there were known algorithms with round complexity $O(\log\log n)$, or $O(\log\log \Delta)$ where $\Delta$ denotes maximum degree, for $1+\epsilon$ approximation of weighted matching and for maximal matching [Czumaj et al., STOC'18, Ghaffari et al. PODC'18; Assadi et al. SODA'19; Behnezhad et al. FOCS'19; Gamlath et al. PODC'19], but these algorithms do not extend to the more general $b$-matching problem., Comment: This paper appeared in Proceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA) 2022

Correlation clustering is a central problem in unsupervised learning, with applications spanning community detection, duplicate detection, automated labelling and many more. In the correlation clustering problem one receives as input a set of nodes and for each node a list of co-clustering preferences, and the goal is to output a clustering that minimizes the disagreement with the specified nodes' preferences. In this paper, we introduce a simple and computationally efficient algorithm for the correlation clustering problem with provable privacy guarantees. Our approximation guarantees are stronger than those shown in prior work and are optimal up to logarithmic factors.

Correlation clustering is a central topic in unsupervised learning, with many applications in ML and data mining. In correlation clustering, one receives as input a signed graph and the goal is to partition it to minimize the number of disagreements. In this work we propose a massively parallel computation (MPC) algorithm for this problem that is considerably faster than prior work. In particular, our algorithm uses machines with memory sublinear in the number of nodes in the graph and returns a constant approximation while running only for a constant number of rounds. To the best of our knowledge, our algorithm is the first that can provably approximate a clustering problem on graphs using only a constant number of MPC rounds in the sublinear memory regime. We complement our analysis with an experimental analysis of our techniques., Comment: ICML 2021 (long talk)

We present a deterministic $(1+\varepsilon)$-approximate maximum matching algorithm in $\mathsf{poly} 1/\varepsilon$ passes in the semi-streaming model, solving the long-standing open problem of breaking the exponential barrier in the dependence on $1/\varepsilon$. Our algorithm exponentially improves on the well-known randomized $(1/\varepsilon)^{O(1/\varepsilon)}$-pass algorithm from the seminal work by McGregor~[APPROX05], the recent deterministic algorithm by Tirodkar with the same pass complexity~[FSTTCS18]. Up to polynomial factors in $1/\varepsilon$, our work matches the state-of-the-art deterministic $(\log n / \log \log n) \cdot (1/\varepsilon)$-pass algorithm by Ahn and Guha~[TOPC18], that is allowed a dependence on the number of nodes $n$. Our result also makes progress on the Open Problem 60 at sublinear.info. Moreover, we design a general framework that simulates our approach for the streaming setting in other models of computation. This framework requires access to an algorithm computing an $O(1)$-approximate maximum matching and an algorithm for processing disjoint $(\mathsf{poly} 1 / \varepsilon)$-size connected components. Instantiating our framework in $\mathsf{CONGEST}$ yields a $\mathsf{poly}(\log{n}, 1/\varepsilon)$ round algorithm for computing $(1+\varepsilon$)-approximate maximum matching. In terms of the dependence on $1/\varepsilon$, this result improves exponentially state-of-the-art result by Lotker, Patt-Shamir, and Pettie~[LPSP15]. Our framework leads to the same quality of improvement in the context of the Massively Parallel Computation model as well.

Interval scheduling is a basic problem in the theory of algorithms and a classical task in combinatorial optimization. We develop a set of techniques for partitioning and grouping jobs based on their starting and ending times, that enable us to view an instance of interval scheduling on many jobs as a union of multiple interval scheduling instances, each containing only a few jobs. Instantiating these techniques in dynamic and local settings of computation leads to several new results. For $(1+\varepsilon)$-approximation of job scheduling of $n$ jobs on a single machine, we develop a fully dynamic algorithm with $O(\frac{\log{n}}{\varepsilon})$ update and $O(\log{n})$ query worst-case time. Further, we design a local computation algorithm that uses only $O(\frac{\log{N}}{\varepsilon})$ queries when all jobs are length at least $1$ and have starting/ending times within $[0,N]$. Our techniques are also applicable in a setting where jobs have rewards/weights. For this case we design a fully dynamic deterministic algorithm whose worst-case update and query time are $\operatorname{poly}(\log n,\frac{1}{\varepsilon})$. Equivalently, this is the first algorithm that maintains a $(1+\varepsilon)$-approximation of the maximum independent set of a collection of weighted intervals in $\operatorname{poly}(\log n,\frac{1}{\varepsilon})$ time updates/queries. This is an exponential improvement in $1/\varepsilon$ over the running time of a randomized algorithm of Henzinger, Neumann, and Wiese ~[SoCG, 2020], while also removing all dependence on the values of the jobs' starting/ending times and rewards, as well as removing the need for any randomness. We also extend our approaches for interval scheduling on a single machine to examine the setting with $M$ machines., Comment: Main result (Theorem 2) has stronger guarantees, updates/queries now in $\operatorname{poly}(\log(n),\frac{1}{\varepsilon})$ time

Submodular maximization has become established as the method of choice for the task of selecting representative and diverse summaries of data. However, if datapoints have sensitive attributes such as gender or age, such machine learning algorithms, left unchecked, are known to exhibit bias: under- or over-representation of particular groups. This has made the design of fair machine learning algorithms increasingly important. In this work we address the question: Is it possible to create fair summaries for massive datasets? To this end, we develop the first streaming approximation algorithms for submodular maximization under fairness constraints, for both monotone and non-monotone functions. We validate our findings empirically on exemplar-based clustering, movie recommendation, DPP-based summarization, and maximum coverage in social networks, showing that fairness constraints do not significantly impact utility., Comment: Accepted to NeurIPS 2020

We consider online algorithms for the {\em page migration problem} that use predictions, potentially imperfect, to improve their performance. The best known online algorithms for this problem, due to Westbrook'94 and Bienkowski et al'17, have competitive ratios strictly bounded away from 1. In contrast, we show that if the algorithm is given a prediction of the input sequence, then it can achieve a competitive ratio that tends to $1$ as the prediction error rate tends to $0$. Specifically, the competitive ratio is equal to $1+O(q)$, where $q$ is the prediction error rate. We also design a ``fallback option'' that ensures that the competitive ratio of the algorithm for {\em any} input sequence is at most $O(1/q)$. Our result adds to the recent body of work that uses machine learning to improve the performance of ``classic'' algorithms.

The task of maximizing a monotone submodular function under a cardinality constraint is at the core of many machine learning and data mining applications, including data summarization, sparse regression and coverage problems. We study this classic problem in the fully dynamic setting, where elements can be both inserted and removed. Our main result is a randomized algorithm that maintains an efficient data structure with a poly-logarithmic amortized update time and yields a $(1/2-\epsilon)$-approximate solution. We complement our theoretical analysis with an empirical study of the performance of our algorithm.

Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing large-scale graphs. By now, we have quite fast algorithms -- usually sublogarithmic-time and often $poly(\log\log n)$-time, or even faster -- for a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widely-adopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an all-to-all manner to process large-scale data. Contributing to this line of work on MPC graph algorithms, we present $poly(\log k) \in poly(\log\log n)$ round MPC algorithms for computing $O(k^{1+{o(1)}})$-spanners in the strongly sublinear regime of local memory. To the best of our knowledge, these are the first sublogarithmic-time MPC algorithms for spanner construction. As primary applications of our spanners, we get two important implications, as follows: -For the MPC setting, we get an $O(\log^2\log n)$-round algorithm for $O(\log^{1+o(1)} n)$ approximation of all pairs shortest paths (APSP) in the near-linear regime of local memory. To the best of our knowledge, this is the first sublogarithmic-time MPC algorithm for distance approximations. -Our result above also extends to the Congested Clique model of distributed computing, with the same round complexity and approximation guarantee. This gives the first sub-logarithmic algorithm for approximating APSP in weighted graphs in the Congested Clique model.

Over the last two decades, frameworks for distributed-memory parallel computation, such as MapReduce, Hadoop, Spark and Dryad, have gained significant popularity with the growing prevalence of large network datasets. The Massively Parallel Computation (MPC) model is the de-facto standard for studying graph algorithms in these frameworks theoretically. Subgraph counting is one such fundamental problem in analyzing massive graphs, with the main algorithmic challenges centering on designing methods which are both scalable and accurate. Given a graph $G=(V, E)$ with $n$ vertices, $m$ edges and $T$ triangles, our first result is an algorithm that outputs a $(1+\varepsilon)$-approximation to $T$, with asymptotically \emph{optimal round and total space complexity} provided any $S \geq \max{(\sqrt m, n^2/m)}$ space per machine and assuming $T=\Omega(\sqrt{m/n})$. Our result gives a quadratic improvement on the bound on $T$ over previous works. We also provide a simple extension of our result to counting \emph{any} subgraph of $k$ size for constant $k \geq 1$. Our second result is an $O_{\varepsilon}(\log \log n)$-round algorithm for exactly counting the number of triangles, whose total space usage is parametrized by the \emph{arboricity} $\alpha$ of the input graph. We extend this result to exactly counting $k$-cliques for any constant $k$. Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri (ITCS 2020) for exactly counting all subgraphs of size at most $5$ can be implemented in the MPC model in total space., Comment: Abstract truncated per arXiv requirements

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 $s^{O(\log{s})} t^{O(\log{s} \cdot (\log \log{s} + \log \log{t}))}$ 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 $(st)^{O(\log{s} \cdot \log{t})}$. 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., Comment: To appear in ACM-SIAM Symposium on Discrete Algorithms (SODA 2020)

Given a source of iid samples of edges of an input graph $G$ with $n$ vertices and $m$ edges, how many samples does one need to compute a constant factor approximation to the maximum matching size in $G$? Moreover, is it possible to obtain such an estimate in a small amount of space? We show that, on the one hand, this problem cannot be solved using a nontrivially sublinear (in $m$) number of samples: $m^{1-o(1)}$ samples are needed. On the other hand, a surprisingly space efficient algorithm for processing the samples exists: $O(\log^2 n)$ bits of space suffice to compute an estimate. Our main technical tool is a new peeling type algorithm for matching that we simulate using a recursive sampling process that crucially ensures that local neighborhood information from `dense' regions of the graph is provided at appropriately higher sampling rates. We show that a delicate balance between exploration depth and sampling rate allows our simulation to not lose precision over a logarithmic number of levels of recursion and achieve a constant factor approximation. The previous best result on matching size estimation from random samples was a $\log^{O(1)} n$ approximation [Kapralov et al'14]. Our algorithm also yields a constant factor approximate local computation algorithm (LCA) for matching with $O(d\log n)$ exploration starting from any vertex. Previous approaches were based on local simulations of randomized greedy, which take $O(d)$ time {\em in expectation over the starting vertex or edge} (Yoshida et al'09, Onak et al'12), and could not achieve a better than $d^2$ runtime. Interestingly, we also show that unlike our algorithm, the local simulation of randomized greedy that is the basis of the most efficient prior results does take $\wt{\Omega}(d^2)\gg O(d\log n)$ time for a worst case edge even for $d=\exp(\Theta(\sqrt{\log n}))$.

We introduce a set of techniques that allow for efficiently generating many independent random walks in the Massive Parallel Computation (MPC) model with space per machine strongly sublinear in the number of vertices. In this space-per-machine regime, many natural approaches to graph problems struggle to overcome the $\Theta(\log n)$ MPC round complexity barrier. Our techniques enable breaking this barrier for PageRank---one of the most important applications of random walks---even in more challenging directed graphs, and for approximate bipartiteness and expansion testing. In the undirected case, we start our random walks from the stationary distribution, which implies that we approximately know the empirical distribution of their next steps. This allows for preparing continuations of random walks in advance and applying a doubling approach. As a result we can generate multiple random walks of length $l$ in $\Theta(\log l)$ rounds on MPC. Moreover, we show that under the popular 1-vs.-2-Cycles conjecture, this round complexity is asymptotically tight. For directed graphs, our approach stems from our treatment of the PageRank Markov chain. We first compute the PageRank for the undirected version of the input graph and then slowly transition towards the directed case, considering convex combinations of the transition matrices in the process. For PageRank, we achieve the following round complexities for damping factor equal to $1 - \epsilon$: * in $O(\log \log n + \log 1 / \epsilon)$ rounds for undirected graphs (with $\tilde O(m / \epsilon^2)$ total space), * in $\tilde O(\log^2 \log n + \log^2 1/\epsilon)$ rounds for directed graphs (with $\tilde O((m+n^{1+o(1)}) / poly\, \epsilon)$ total space).

We propose the first adversarially robust algorithm for monotone submodular maximization under single and multiple knapsack constraints with scalable implementations in distributed and streaming settings. For a single knapsack constraint, our algorithm outputs a robust summary of almost optimal (up to polylogarithmic factors) size, from which a constant-factor approximation to the optimal solution can be constructed. For multiple knapsack constraints, our approximation is within a constant-factor of the best known non-robust solution. We evaluate the performance of our algorithms by comparison to natural robustifications of existing non-robust algorithms under two objectives: 1) dominating set for large social network graphs from Facebook and Twitter collected by the Stanford Network Analysis Project (SNAP), 2) movie recommendations on a dataset from MovieLens. Experimental results show that our algorithms give the best objective for a majority of the inputs and show strong performance even compared to offline algorithms that are given the set of removals in advance., Comment: To appear in KDD 2019

A non-redundant integer cone generator (NICG) of dimension $d$ is a set $S$ of vectors from $\{0,1\}^d$ whose vector sum cannot be generated as a positive integer linear combination of a proper subset of $S$. The largest possible cardinality of NICG of a dimension $d$, denoted by $N(d)$, provides an upper bound on the sparsity of systems of integer equations with a large number of integer variables. A better estimate of $N(d)$ means that we can consider smaller sub-systems of integer equations when solving systems with many integer variables. Furthermore, given that we can reduce constraints on set algebra expressions to constraints on cardinalities of Venn regions, tighter upper bound on $N(d)$ yields a more efficient decision procedure for a logic of sets with cardinality constraints (BAPA), which has been applied in software verification. Previous attempts to compute $N(d)$ using SAT solvers have not succeeded even for $d=3$. The only known values were computed manually: $N(d)=d$ for $d < 4$ and $N(4) > 4$. We provide the first exact values for $d > 3$, namely, $N(4)=5$, $N(5)=7$, and $N(6)=9$, which is a significant improvement of the known asymptotic bound (which would give only e.g. $N(6) \le 29$, making a decision procedure impractical for $d=6$). We also give lower bounds for $N(7)$, $N(8)$, $N(9)$, and $N(10)$, which are: $11$, $13$, $14$, and $16$, respectively. We describe increasingly sophisticated specialized search algorithms that we used to explore the space of non-redundant generators and obtain these results.

We design a generic method for reducing the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the streaming model and in the massively parallel computation (MPC) model. In the context of streaming with random edge arrivals, our techniques yield a $(1/2+c)$-approximation algorithm thus breaking the natural barrier of $1/2$. For multi-pass streaming and the MPC model, we show that any algorithm computing a $(1-\delta)$-approximate unweighted matching in bipartite graphs can be translated into an algorithm that computes a $(1-\varepsilon(\delta))$-approximate maximum weighted matching. Furthermore, this translation incurs only a constant factor (that depends on $\varepsilon> 0$) overhead in the complexity. Instantiating this with the current best multi-pass streaming and MPC algorithms for unweighted matchings yields the following results for maximum weighted matchings: * A $(1-\varepsilon)$-approximation streaming algorithm that uses $O_\varepsilon(1)$ passes and $O_\varepsilon(n\, \text{poly} (\log n))$ memory. This is the first $(1-\varepsilon)$-approximation streaming algorithm for weighted matchings that uses a constant number of passes (only depending on $\varepsilon$). * A $(1 - \varepsilon)$-approximation algorithm in the MPC model that uses $O_\varepsilon(\log \log n)$ rounds, $O(m/n)$ machines per round, and $O_\varepsilon(n\, \text{poly}(\log n))$ memory per machine. This improves upon the previous best approximation guarantee of $(1/2-\varepsilon)$ for weighted graphs.

Many tasks in machine learning and data mining, such as data diversification, non-parametric learning, kernel machines, clustering etc., require extracting a small but representative summary from a massive dataset. Often, such problems can be posed as maximizing a submodular set function subject to a cardinality constraint. We consider this question in the streaming setting, where elements arrive over time at a fast pace and thus we need to design an efficient, low-memory algorithm. One such method, proposed by Badanidiyuru et al. (2014), always finds a $0.5$-approximate solution. Can this approximation factor be improved? We answer this question affirmatively by designing a new algorithm SALSA for streaming submodular maximization. It is the first low-memory, single-pass algorithm that improves the factor $0.5$, under the natural assumption that elements arrive in a random order. We also show that this assumption is necessary, i.e., that there is no such algorithm with better than $0.5$-approximation when elements arrive in arbitrary order. Our experiments demonstrate that SALSA significantly outperforms the state of the art in applications related to exemplar-based clustering, social graph analysis, and recommender systems.

We present $O(\log\log n)$-round algorithms in the Massively Parallel Computation (MPC) model, with $\tilde{O}(n)$ memory per machine, that compute a maximal independent set, a $1+\epsilon$ approximation of maximum matching, and a $2+\epsilon$ approximation of minimum vertex cover, for any $n$-vertex graph and any constant $\epsilon>0$. These improve the state of the art as follows: - Our MIS algorithm leads to a simple $O(\log\log \Delta)$-round MIS algorithm in the Congested Clique model of distributed computing, which improves on the $\tilde{O}(\sqrt{\log \Delta})$-round algorithm of Ghaffari [PODC'17]. - Our $O(\log\log n)$-round $(1+\epsilon)$-approximate maximum matching algorithm simplifies or improves on the following prior work: $O(\log^2\log n)$-round $(1+\epsilon)$-approximation algorithm of Czumaj et al. [STOC'18] and $O(\log\log n)$-round $(1+\epsilon)$-approximation algorithm of Assadi et al. [SODA'19]. - Our $O(\log\log n)$-round $(2+\epsilon)$-approximate minimum vertex cover algorithm improves on an $O(\log\log n)$-round $O(1)$-approximation of Assadi et al. [arXiv'17].

Sparsity-based methods are widely used in machine learning, statistics, and signal processing. There is now a rich class of structured sparsity approaches that expand the modeling power of the sparsity paradigm and incorporate constraints such as group sparsity, graph sparsity, or hierarchical sparsity. While these sparsity models offer improved sample complexity and better interpretability, the improvements come at a computational cost: it is often challenging to optimize over the (non-convex) constraint sets that capture various sparsity structures. In this paper, we make progress in this direction in the context of separated sparsity -- a fundamental sparsity notion that captures exclusion constraints in linearly ordered data such as time series. While prior algorithms for computing a projection onto this constraint set required quadratic time, we provide a perturbed Lagrangian relaxation approach that computes provably exact projection in only nearly-linear time. Although the sparsity constraint is non-convex, our perturbed Lagrangian approach is still guaranteed to find a globally optimal solution. In experiments, our new algorithms offer a 10$\times$ speed-up already on moderately-size inputs.

We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm's memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-T. It is based on a novel partitioning structure and an exponentially decreasing thresholding rule. STAR-T makes one pass over the data and retains a short but robust summary. We show that after the removal of any m elements from the obtained summary, a simple greedy algorithm STAR-T-GREEDY that runs on the remaining elements achieves a constant-factor approximation guarantee. In two different data summarization tasks, we demonstrate that it matches or outperforms existing greedy and streaming methods, even if they are allowed the benefit of knowing the removed subset in advance., Comment: To appear in NIPS 2017

For over a decade now we have been witnessing the success of {\em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms? A prominent example here is the {\em maximum matching} problem---one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in $O(\log{n})$ rounds. However, the exact complexity of this problem in the MPC framework is still far from understood. Lattanzi et al. showed that if each machine has $n^{1+\Omega(1)}$ memory, this problem can also be solved $2$-approximately in a constant number of rounds. These techniques, as well as the approaches developed in the follow up work, seem though to get stuck in a fundamental way at roughly $O(\log{n})$ rounds once we enter the near-linear memory regime. It is thus entirely possible that in this regime, which captures in particular the case of sparse graph computations, the best MPC round complexity matches what one can already get in the PRAM model, without the need to take advantage of the extra local computation power. In this paper, we finally refute that perplexing possibility. That is, we break the above $O(\log n)$ round complexity bound even in the case of {\em slightly sublinear} memory per machine. In fact, our improvement here is {\em almost exponential}: we are able to deliver a $(2+\epsilon)$-approximation to maximum matching, for any fixed constant $\epsilon>0$, in $O((\log \log n)^2)$ rounds.

We study the problem of maximizing a monotone submodular function subject to a cardinality constraint $k$, with the added twist that a number of items $\tau$ from the returned set may be removed. We focus on the worst-case setting considered in (Orlin et al., 2016), in which a constant-factor approximation guarantee was given for $\tau = o(\sqrt{k})$. In this paper, we solve a key open problem raised therein, presenting a new Partitioned Robust (PRo) submodular maximization algorithm that achieves the same guarantee for more general $\tau = o(k)$. Our algorithm constructs partitions consisting of buckets with exponentially increasing sizes, and applies standard submodular optimization subroutines on the buckets in order to construct the robust solution. We numerically demonstrate the performance of PRo in data summarization and influence maximization, demonstrating gains over both the greedy algorithm and the algorithm of (Orlin et al., 2016)., Comment: Accepted to ICML 2017

Photodetectors are typically based on photocurrent generation from electron-hole pairs in semiconductor structures and on bolometry for wavelengths that are below bandgap absorption. In both cases, resonant plasmonic and nanophotonic structures have been successfully used to enhance performance. In this work, we demonstrate subwavelength thermoelectric nanostructures designed for resonant spectrally selective absorption, which creates large enough localized temperature gradients to generate easily measureable thermoelectric voltages. We show that such structures are tunable and are capable of highly wavelength specific detection, with an input power responsivity of up to 119 V/W (referenced to incident illumination), and response times of nearly 3 kHz, by combining resonant absorption and thermoelectric junctions within a single structure, yielding a bandgap-independent photodetection mechanism. We report results for both resonant nanophotonic bismuth telluride-antimony telluride structures and chromel-alumel structures as examples of a broad class of nanophotonic thermoelectric structures useful for fast, low-cost and robust optoelectronic applications such as non-bandgap-limited hyperspectral and broad-band photodetectors.

We present and study the Static-Routing-Resiliency problem, motivated by routing on the Internet: Given a graph $G$, a unique destination vertex $d$, and an integer constant $c>0$, does there exist a static and destination-based routing scheme such that the correct delivery of packets from any source $s$ to the destination $d$ is guaranteed so long as (1) no more than $c$ edges fail and (2) there exists a physical path from $s$ to $d$? We embark upon a systematic exploration of this fundamental question in a variety of models (deterministic routing, randomized routing, with packet-duplication, with packet-header-rewriting) and present both positive and negative results that relate the edge-connectivity of a graph, i.e., the minimum number of edges whose deletion partitions $G$, to its resiliency., Comment: 30 pages

Let $G = (V,E)$ denote a simple graph with the vertex set $V$ and the edge set $E$. The profile of a vertex set $V'\subseteq V$ denotes the multiset of pairwise distances between the vertices of $V'$. Two disjoint subsets of $V$ are \emph{homometric}, if their profiles are the same. If $G$ is a tree on $n$ vertices we prove that its vertex sets contains a pair of disjoint homometric subsets of size at least $\sqrt{n/2} - 1$. Previously it was known that such a pair of size at least roughly $n^{1/3}$ exists. We get a better result in case of haircomb trees, in which we are able to find a pair of disjoint homometric sets of size at least $cn^{2/3}$ for a constant $c > 0$.