arXiv subjects

Sepehr Assadi

Publications and source records attributed to Sepehr Assadi.

At least 73 records · Page 4Linked to original sources

arXiv2020

When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear-Time

Maximal independent set (MIS), maximal matching (MM), and $(Δ+1)$-coloring in graphs of maximum degree $Δ$ are among the most prominent algorithmic graph theory problems. They are all solvable by a simple linear-time greedy algorithm and up until very recently this constituted the state-of-the-art. In SODA 2019, Assadi, Chen, and Khanna gave a randomized algorithm for $(Δ+1)$-coloring that runs in $\widetilde{O}(n\sqrt{n})$ time, which even for moderately dense graphs is sublinear in the input size. The work of Assadi et al. however contained a spoiler for MIS and MM: neither problems provably admits a sublinear-time algorithm in general graphs. In this work, we dig deeper into the possibility of achieving sublinear-time algorithms for MIS and MM. The neighborhood independence number of a graph $G$, denoted by $β(G)$, is the size of the largest independent set in the neighborhood of any vertex. We identify $β(G)$ as the ``right'' parameter to measure the runtime of MIS and MM algorithms: Although graphs of bounded neighborhood independence may be very dense (clique is one example), we prove that carefully chosen variants of greedy algorithms for MIS and MM run in $O(nβ(G))$ and $O(n\log{n}\cdotβ(G))$ time respectively on any $n$-vertex graph $G$. We complement this positive result by observing that a simple extension of the lower bound of Assadi et.al. implies that $Ω(nβ(G))$ time is also necessary for any algorithm to either problem for all values of $β(G)$ from $1$ to $Θ(n)$. We note that our algorithm for MIS is deterministic while for MM we use randomization which we prove is unavoidable: any deterministic algorithm for MM requires $Ω(n^2)$ time even for $β(G) = 2$.

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arXiv2019

Improved Truthful Mechanisms for Combinatorial Auctions with Submodular Bidders

A longstanding open problem in Algorithmic Mechanism Design is to design computationally-efficient truthful mechanisms for (approximately) maximizing welfare in combinatorial auctions with submodular bidders. The first such mechanism was obtained by Dobzinski, Nisan, and Schapira [STOC'06] who gave an $O(\log^2{m})$-approximation where $m$ is the number of items. This problem has been studied extensively since, culminating in an $O(\sqrt{\log{m}})$-approximation mechanism by Dobzinski [STOC'16]. We present a computationally-efficient truthful mechanism with approximation ratio that improves upon the state-of-the-art by an exponential factor. In particular, our mechanism achieves an $O((\log\log{m})^3)$-approximation in expectation, uses only $O(n)$ demand queries, and has universal truthfulness guarantee. This settles an open question of Dobzinski on whether $Θ(\sqrt{\log{m}})$ is the best approximation ratio in this setting in negative.

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arXiv2019

Distributed Weighted Matching via Randomized Composable Coresets

Maximum weight matching is one of the most fundamental combinatorial optimization problems with a wide range of applications in data mining and bioinformatics. Developing distributed weighted matching algorithms is challenging due to the sequential nature of efficient algorithms for this problem. In this paper, we develop a simple distributed algorithm for the problem on general graphs with approximation guarantee of $2+\varepsilon$ that (nearly) matches that of the sequential greedy algorithm. A key advantage of this algorithm is that it can be easily implemented in only two rounds of computation in modern parallel computation frameworks such as MapReduce. We also demonstrate the efficiency of our algorithm in practice on various graphs (some with half a trillion edges) by achieving objective values always close to what is achievable in the centralized setting.

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arXiv2019

Polynomial Pass Lower Bounds for Graph Streaming Algorithms

We present new lower bounds that show that a polynomial number of passes are necessary for solving some fundamental graph problems in the streaming model of computation. For instance, we show that any streaming algorithm that finds a weighted minimum $s$-$t$ cut in an $n$-vertex undirected graph requires $n^{2-o(1)}$ space unless it makes $n^{Ω(1)}$ passes over the stream. To prove our lower bounds, we introduce and analyze a new four-player communication problem that we refer to as the hidden-pointer chasing problem. This is a problem in spirit of the standard pointer chasing problem with the key difference that the pointers in this problem are hidden to players and finding each one of them requires solving another communication problem, namely the set intersection problem. Our lower bounds for graph problems are then obtained by reductions from the hidden-pointer chasing problem. Our hidden-pointer chasing problem appears flexible enough to find other applications and is therefore interesting in its own right. To showcase this, we further present an interesting application of this problem beyond streaming algorithms. Using a reduction from hidden-pointer chasing, we prove that any algorithm for submodular function minimization needs to make $n^{2-o(1)}$ value queries to the function unless it has a polynomial degree of adaptivity.

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arXiv2019

Distributed and Streaming Linear Programming in Low Dimensions

We study linear programming and general LP-type problems in several big data (streaming and distributed) models. We mainly focus on low dimensional problems in which the number of constraints is much larger than the number of variables. Low dimensional LP-type problems appear frequently in various machine learning tasks such as robust regression, support vector machines, and core vector machines. As supporting large-scale machine learning queries in database systems has become an important direction for database research, obtaining efficient algorithms for low dimensional LP-type problems on massive datasets is of great value. In this paper we give both upper and lower bounds for LP-type problems in distributed and streaming models. Our bounds are almost tight when the dimensionality of the problem is a fixed constant.

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arXiv2019

Sublinear Algorithms for $(Δ+ 1)$ Vertex Coloring

Any graph with maximum degree $Δ$ admits a proper vertex coloring with $Δ+ 1$ colors that can be found via a simple sequential greedy algorithm in linear time and space. But can one find such a coloring via a sublinear algorithm? We answer this fundamental question in the affirmative for several canonical classes of sublinear algorithms including graph streaming, sublinear time, and massively parallel computation (MPC) algorithms. In particular, we design: * A single-pass semi-streaming algorithm in dynamic streams using $\tilde{O}(n)$ space. The only known semi-streaming algorithm prior to our work was a folklore O(log n)-pass algorithm obtained by simulating classical distributed algorithms in the streaming model. * A sublinear-time algorithm in the standard query model that allows neighbor queries and pair queries using $\tilde{O}(n\sqrt{n})$ time. We further show that any algorithm that outputs a valid coloring with sufficiently large constant probability requires $Ω(n\sqrt{n})$ time. No non-trivial sublinear time algorithms were known prior to our work. * A parallel algorithm in the massively parallel computation (MPC) model using $\tilde{O}(n)$ memory per machine and $O(1)$ MPC rounds. Our number of rounds significantly improves upon the recent $O(\log\logΔ\cdot\log^*{(n)})$-round algorithm of Parter [ICALP 2018]. At the core of our results is a remarkably simple meta-algorithm for the $(Δ+1)$ coloring problem: Sample $O(\log{n})$ colors for each vertex from the $Δ+1$ colors; find a proper coloring of the graph using only the sampled colors. We prove that the sampled set of colors with high probability contains a proper coloring of the input graph. The sublinear algorithms are then obtained by designing efficient algorithms for finding a proper coloring of the graph from the sampled colors in the corresponding models.

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arXiv2018

Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs

As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems, such as maximum matching and minimum vertex cover, on large datasets. For massive inputs, several different computational models have been introduced, including the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In each model, algorithms are analyzed in terms of resources such as space used or rounds of communication needed, in addition to the more traditional approximation ratio. In this paper, we give a single unified approach that yields better approximation algorithms for matching and vertex cover in all these models. The highlights include: * The first one pass, significantly-better-than-2-approximation for matching in random arrival streams that uses subquadratic space, namely a $(1.5+ε)$-approximation streaming algorithm that uses $O(n^{1.5})$ space for constant $ε> 0$. * The first 2-round, better-than-2-approximation for matching in the MPC model that uses subquadratic space per machine, namely a $(1.5+ε)$-approximation algorithm with $O(\sqrt{mn} + n)$ memory per machine for constant $ε> 0$. By building on our unified approach, we further develop parallel algorithms in the MPC model that give a $(1 + ε)$-approximation to matching and an $O(1)$-approximation to vertex cover in only $O(\log\log{n})$ MPC rounds and $O(n/poly\log{(n)})$ memory per machine. These results settle multiple open questions posed in the recent paper of Czumaj~et.al. [STOC 2018].

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arXiv2018

A Simple Sublinear-Time Algorithm for Counting Arbitrary Subgraphs via Edge Sampling

In the subgraph counting problem, we are given a input graph $G(V, E)$ and a target graph $H$; the goal is to estimate the number of occurrences of $H$ in $G$. Our focus here is on designing sublinear-time algorithms for approximately counting occurrences of $H$ in $G$ in the setting where the algorithm is given query access to $G$. This problem has been studied in several recent papers which primarily focused on specific families of graphs $H$ such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs $H$. This is in sharp contrast to the closely related subgraph enumeration problem that has received significant attention in the database community as the database join problem. The AGM bound shows that the maximum number of occurrences of any arbitrary subgraph $H$ in a graph $G$ with $m$ edges is $O(m^{ρ(H)})$, where $ρ(H)$ is the fractional edge-cover of $H$, and enumeration algorithms with matching runtime are known for any $H$. We bridge this gap between subgraph counting and subgraph enumeration by designing a sublinear-time algorithm that can estimate the number of any arbitrary subgraph $H$ in $G$, denoted by $\#H$, to within a $(1\pm ε)$-approximation w.h.p. in $O(\frac{m^{ρ(H)}}{\#H}) \cdot poly(\log{n},1/ε)$ time. Our algorithm is allowed the standard set of queries for general graphs, namely degree queries, pair queries and neighbor queries, plus an additional edge-sample query that returns an edge chosen uniformly at random. The performance of our algorithm matches those of Eden et.al. [FOCS 2015, STOC 2018] for counting triangles and cliques and extend them to all choices of subgraph $H$ under the additional assumption of edge-sample queries. We further show that our algorithm works for the more general database join size estimation problem and prove a matching lower bound for this problem.

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arXiv2018

Secretary Ranking with Minimal Inversions

We study a twist on the classic secretary problem, which we term the secretary ranking problem: elements from an ordered set arrive in random order and instead of picking the maximum element, the algorithm is asked to assign a rank, or position, to each of the elements. The rank assigned is irrevocable and is given knowing only the pairwise comparisons with elements previously arrived. The goal is to minimize the distance of the rank produced to the true rank of the elements measured by the Kendall-Tau distance, which corresponds to the number of pairs that are inverted with respect to the true order. Our main result is a matching upper and lower bound for the secretary ranking problem. We present an algorithm that ranks $n$ elements with only $O(n^{3/2})$ inversions in expectation, and show that any algorithm necessarily suffers $Ω(n^{3/2})$ inversions when there are $n$ available positions. In terms of techniques, the analysis of our algorithm draws connections to linear probing in the hashing literature, while our lower bound result relies on a general anti-concentration bound for a generic balls and bins sampling process. We also consider the case where the number of positions $m$ can be larger than the number of secretaries $n$ and provide an improved bound by showing a connection of this problem with random binary trees.

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arXiv2018

Towards a Unified Theory of Sparsification for Matching Problems

In this paper, we present a construction of a `matching sparsifier', that is, a sparse subgraph of the given graph that preserves large matchings approximately and is robust to modifications of the graph. We use this matching sparsifier to obtain several new algorithmic results for the maximum matching problem: * An almost $(3/2)$-approximation one-way communication protocol for the maximum matching problem, significantly simplifying the $(3/2)$-approximation protocol of Goel, Kapralov, and Khanna (SODA 2012) and extending it from bipartite graphs to general graphs. * An almost $(3/2)$-approximation algorithm for the stochastic matching problem, improving upon and significantly simplifying the previous $1.999$-approximation algorithm of Assadi, Khanna, and Li (EC 2017). * An almost $(3/2)$-approximation algorithm for the fault-tolerant matching problem, which, to our knowledge, is the first non-trivial algorithm for this problem. Our matching sparsifier is obtained by proving new properties of the edge-degree constrained subgraph (EDCS) of Bernstein and Stein (ICALP 2015; SODA 2016)---designed in the context of maintaining matchings in dynamic graphs---that identifies EDCS as an excellent choice for a matching sparsifier. This leads to surprisingly simple and non-technical proofs of the above results in a unified way. Along the way, we also provide a much simpler proof of the fact that an EDCS is guaranteed to contain a large matching, which may be of independent interest.

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arXiv2018

Stochastic Submodular Cover with Limited Adaptivity

In the submodular cover problem, we are given a non-negative monotone submodular function $f$ over a ground set $E$ of items, and the goal is to choose a smallest subset $S \subseteq E$ such that $f(S) = Q$ where $Q = f(E)$. In the stochastic version of the problem, we are given $m$ stochastic items which are different random variables that independently realize to some item in $E$, and the goal is to find a smallest set of stochastic items whose realization $R$ satisfies $f(R) = Q$. The problem captures as a special case the stochastic set cover problem and more generally, stochastic covering integer programs. We define an $r$-round adaptive algorithm to be an algorithm that chooses a permutation of all available items in each round $k \in [r]$, and a threshold $τ_k$, and realizes items in the order specified by the permutation until the function value is at least $τ_k$. The permutation for each round $k$ is chosen adaptively based on the realization in the previous rounds, but the ordering inside each round remains fixed regardless of the realizations seen inside the round. Our main result is that for any integer $r$, there exists a poly-time $r$-round adaptive algorithm for stochastic submodular cover whose expected cost is $\tilde{O}(Q^{{1}/{r}})$ times the expected cost of a fully adaptive algorithm. Prior to our work, such a result was not known even for the case of $r=1$ and when $f$ is the coverage function. On the other hand, we show that for any $r$, there exist instances of the stochastic submodular cover problem where no $r$-round adaptive algorithm can achieve better than $Ω(Q^{{1}/{r}})$ approximation to the expected cost of a fully adaptive algorithm. Our lower bound result holds even for coverage function and for algorithms with unbounded computational power.

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arXiv2018

Tight Bounds on the Round Complexity of the Distributed Maximum Coverage Problem

We study the maximum $k$-set coverage problem in the following distributed setting. A collection of sets $S_1,\ldots,S_m$ over a universe $[n]$ is partitioned across $p$ machines and the goal is to find $k$ sets whose union covers the most number of elements. The computation proceeds in synchronous rounds. In each round, all machines simultaneously send a message to a central coordinator who then communicates back to all machines a summary to guide the computation for the next round. At the end, the coordinator outputs the answer. The main measures of efficiency in this setting are the approximation ratio of the returned solution, the communication cost of each machine, and the number of rounds of computation. Our main result is an asymptotically tight bound on the tradeoff between these measures for the distributed maximum coverage problem. We first show that any $r$-round protocol for this problem either incurs a communication cost of $ k \cdot m^{Ω(1/r)}$ or only achieves an approximation factor of $k^{Ω(1/r)}$. This implies that any protocol that simultaneously achieves good approximation ratio ($O(1)$ approximation) and good communication cost ($\widetilde{O}(n)$ communication per machine), essentially requires logarithmic (in $k$) number of rounds. We complement our lower bound result by showing that there exist an $r$-round protocol that achieves an $\frac{e}{e-1}$-approximation (essentially best possible) with a communication cost of $k \cdot m^{O(1/r)}$ as well as an $r$-round protocol that achieves a $k^{O(1/r)}$-approximation with only $\widetilde{O}(n)$ communication per each machine (essentially best possible). We further use our results in this distributed setting to obtain new bounds for the maximum coverage problem in two other main models of computation for massive datasets, namely, the dynamic streaming model and the MapReduce model.

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arXiv2018

Fully Dynamic Maximal Independent Set with Sublinear in n Update Time

The first fully dynamic algorithm for maintaining a maximal independent set (MIS) with update time that is sublinear in the number of edges was presented recently by the authors of this paper [Assadi et.al. STOC'18]. The algorithm is deterministic and its update time is $O(m^{3/4})$, where $m$ is the (dynamically changing) number of edges. Subsequently, Gupta and Khan and independently Du and Zhang [arXiv, April 2018] presented deterministic algorithms for dynamic MIS with update times of $O(m^{2/3})$ and $O(m^{2/3} \sqrt{\log m})$, respectively. Du and Zhang also gave a randomized algorithm with update time $\widetilde{O}(\sqrt{m})$. Moreover, they provided some partial (conditional) hardness results hinting that update time of $m^{1/2-ε}$, and in particular $n^{1-ε}$ for $n$-vertex dense graphs, is a natural barrier for this problem for any constant $ε>0$, for both deterministic and randomized algorithms that satisfy a certain natural property. In this paper, we break this natural barrier and present the first fully dynamic (randomized) algorithm for maintaining an MIS with update time that is always sublinear in the number of vertices, namely, an $\widetilde{O}(\sqrt{n})$ expected amortized update time algorithm. We also show that a simpler variant of our algorithm can already achieve an $\widetilde{O}(m^{1/3})$ expected amortized update time, which results in an improved performance over our $\widetilde{O}(\sqrt{n})$ update time algorithm for sufficiently sparse graphs, and breaks the $m^{1/2}$ barrier of Du and Zhang for all values of $m$.

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arXiv2018

Massively Parallel Algorithms for Finding Well-Connected Components in Sparse Graphs

A fundamental question that shrouds the emergence of massively parallel computing (MPC) platforms is how can the additional power of the MPC paradigm be leveraged to achieve faster algorithms compared to classical parallel models such as PRAM? Previous research has identified the sparse graph connectivity problem as a major obstacle to such improvement: While classical logarithmic-round PRAM algorithms for finding connected components in any $n$-vertex graph have been known for more than three decades, no $o(\log{n})$-round MPC algorithms are known for this task with truly sublinear in $n$ memory per machine. This problem arises when processing massive yet sparse graphs with $O(n)$ edges, for which the interesting setting of parameters is $n^{1-Ω(1)}$ memory per machine. It is conjectured that achieving an $o(\log{n})$-round algorithm for connectivity on general sparse graphs with $n^{1-Ω(1)}$ per-machine memory may not be possible, and this conjecture also forms the basis for multiple conditional hardness results on the round complexity of other problems in the MPC model. We take an opportunistic approach towards the sparse graph connectivity problem, by designing an algorithm with improved performance guarantees in terms of the connectivity structure of the input graph. Formally, we design an algorithm that finds all connected components with spectral gap at least $λ$ in a graph in $O(\log\log{n} + \log{(1/λ)})$ MPC rounds and $n^{Ω(1)}$ memory per machine. As such, this algorithm achieves an exponential round reduction on sparse "well-connected" components (i.e., $λ\geq 1/\text{polylog}{(n)}$) using only $n^{Ω(1)}$ memory per machine and $\widetilde{O}(n)$ total memory, and still operates in $o(\log n)$ rounds even when $λ= 1/n^{o(1)}$.

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arXiv2018

Fully Dynamic Maximal Independent Set with Sublinear Update Time

A maximal independent set (MIS) can be maintained in an evolving $m$-edge graph by simply recomputing it from scratch in $O(m)$ time after each update. But can it be maintained in time sublinear in $m$ in fully dynamic graphs? We answer this fundamental open question in the affirmative. We present a deterministic algorithm with amortized update time $O(\min\{Δ,m^{3/4}\})$, where $Δ$ is a fixed bound on the maximum degree in the graph and $m$ is the (dynamically changing) number of edges. We further present a distributed implementation of our algorithm with $O(\min\{Δ,m^{3/4}\})$ amortized message complexity, and $O(1)$ amortized round complexity and adjustment complexity (the number of vertices that change their output after each update). This strengthens a similar result by Censor-Hillel, Haramaty, and Karnin (PODC'16) that required an assumption of a non-adaptive oblivious adversary.

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arXiv2017

Simple Round Compression for Parallel Vertex Cover

Recently, Czumaj et.al. (arXiv 2017) presented a parallel (almost) $2$-approximation algorithm for the maximum matching problem in only $O({(\log\log{n})^2})$ rounds of the massive parallel computation (MPC) framework, when the memory per machine is $O(n)$. The main approach in their work is a way of compressing $O(\log{n})$ rounds of a distributed algorithm for maximum matching into only $O({(\log\log{n})^2})$ MPC rounds. In this note, we present a similar algorithm for the closely related problem of approximating the minimum vertex cover in the MPC framework. We show that one can achieve an $O(\log{n})$ approximation to minimum vertex cover in only $O(\log\log{n})$ MPC rounds when the memory per machine is $O(n)$. Our algorithm for vertex cover is similar to the maximum matching algorithm of Czumaj et.al. but avoids many of the intricacies in their approach and as a result admits a considerably simpler analysis (at a cost of a worse approximation guarantee). We obtain this result by modifying a previous parallel algorithm by Khanna and the author (SPAA 2017) for vertex cover that allowed for compressing $O(\log{n})$ rounds of a distributed algorithm into constant MPC rounds when the memory allowed per machine is $O(n\sqrt{n})$.

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arXiv2017

Randomized Composable Coresets for Matching and Vertex Cover

A common approach for designing scalable algorithms for massive data sets is to distribute the computation across, say $k$, machines and process the data using limited communication between them. A particularly appealing framework here is the simultaneous communication model whereby each machine constructs a small representative summary of its own data and one obtains an approximate/exact solution from the union of the representative summaries. If the representative summaries needed for a problem are small, then this results in a communication-efficient and round-optimal protocol. While many fundamental graph problems admit efficient solutions in this model, two prominent problems are notably absent from the list of successes, namely, the maximum matching problem and the minimum vertex cover problem. Indeed, it was shown recently that for both these problems, even achieving a polylog$(n)$ approximation requires essentially sending the entire input graph from each machine. The main insight of our work is that the intractability of matching and vertex cover in the simultaneous communication model is inherently connected to an adversarial partitioning of the underlying graph across machines. We show that when the underlying graph is randomly partitioned across machines, both these problems admit randomized composable coresets of size $\widetilde{O}(n)$ that yield an $\widetilde{O}(1)$-approximate solution. This results in an $\widetilde{O}(1)$-approximation simultaneous protocol for these problems with $\widetilde{O}(nk)$ total communication when the input is randomly partitioned across $k$ machines. We further prove the optimality of our results. Finally, by a standard application of composable coresets, our results also imply MapReduce algorithms with the same approximation guarantee in one or two rounds of communication

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arXiv2017

The Stochastic Matching Problem: Beating Half with a Non-Adaptive Algorithm

In the stochastic matching problem, we are given a general (not necessarily bipartite) graph $G(V,E)$, where each edge in $E$ is realized with some constant probability $p > 0$ and the goal is to compute a bounded-degree (bounded by a function depending only on $p$) subgraph $H$ of $G$ such that the expected maximum matching size in $H$ is close to the expected maximum matching size in $G$. The algorithms in this setting are considered non-adaptive as they have to choose the subgraph $H$ without knowing any information about the set of realized edges in $G$. Originally motivated by an application to kidney exchange, the stochastic matching problem and its variants have received significant attention in recent years. The state-of-the-art non-adaptive algorithms for stochastic matching achieve an approximation ratio of $\frac{1}{2}-ε$ for any $ε> 0$, naturally raising the question that if $1/2$ is the limit of what can be achieved with a non-adaptive algorithm. In this work, we resolve this question by presenting the first algorithm for stochastic matching with an approximation guarantee that is strictly better than $1/2$: the algorithm computes a subgraph $H$ of $G$ with the maximum degree $O(\frac{\log{(1/ p)}}{p})$ such that the ratio of expected size of a maximum matching in realizations of $H$ and $G$ is at least $1/2+δ_0$ for some absolute constant $δ_0 > 0$. The degree bound on $H$ achieved by our algorithm is essentially the best possible (up to an $O(\log{(1/p)})$ factor) for any constant factor approximation algorithm, since an $Ω(\frac{1}{p})$ degree in $H$ is necessary for a vertex to acquire at least one incident edge in a realization.

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