30,447 research outputs found

    New deterministic approximation algorithms for fully dynamic matching

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    We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a (2+ϵ)(2+\epsilon)-approximate maximum matching in general graphs with O(poly(logn,1/ϵ))O(\text{poly}(\log n, 1/\epsilon)) update time. (2) An algorithm that maintains an αK\alpha_K approximation of the {\em value} of the maximum matching with O(n2/K)O(n^{2/K}) update time in bipartite graphs, for every sufficiently large constant positive integer KK. Here, 1αK<21\leq \alpha_K < 2 is a constant determined by the value of KK. Result (1) is the first deterministic algorithm that can maintain an o(logn)o(\log n)-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best {\em randomized} polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with {\em arbitrarily small polynomial} update time on bipartite graphs. Previously the best update time for this problem was O(m1/4)O(m^{1/4}) [Bernstein et al. ICALP 2015], where mm is the current number of edges in the graph.Comment: To appear in STOC 201

    On Regularity Lemma and Barriers in Streaming and Dynamic Matching

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    We present a new approach for finding matchings in dense graphs by building on Szemer\'edi's celebrated Regularity Lemma. This allows us to obtain non-trivial albeit slight improvements over longstanding bounds for matchings in streaming and dynamic graphs. In particular, we establish the following results for nn-vertex graphs: * A deterministic single-pass streaming algorithm that finds a (1o(1))(1-o(1))-approximate matching in o(n2)o(n^2) bits of space. This constitutes the first single-pass algorithm for this problem in sublinear space that improves over the 12\frac{1}{2}-approximation of the greedy algorithm. * A randomized fully dynamic algorithm that with high probability maintains a (1o(1))(1-o(1))-approximate matching in o(n)o(n) worst-case update time per each edge insertion or deletion. The algorithm works even against an adaptive adversary. This is the first o(n)o(n) update-time dynamic algorithm with approximation guarantee arbitrarily close to one. Given the use of regularity lemma, the improvement obtained by our algorithms over trivial bounds is only by some (logn)Θ(1)(\log^*{n})^{\Theta(1)} factor. Nevertheless, in each case, they show that the ``right'' answer to the problem is not what is dictated by the previous bounds. Finally, in the streaming model, we also present a randomized (1o(1))(1-o(1))-approximation algorithm whose space can be upper bounded by the density of certain Ruzsa-Szemer\'edi (RS) graphs. While RS graphs by now have been used extensively to prove streaming lower bounds, ours is the first to use them as an upper bound tool for designing improved streaming algorithms

    Fully Dynamic Matching in Bipartite Graphs

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    Maximum cardinality matching in bipartite graphs is an important and well-studied problem. The fully dynamic version, in which edges are inserted and deleted over time has also been the subject of much attention. Existing algorithms for dynamic matching (in general graphs) seem to fall into two groups: there are fast (mostly randomized) algorithms that do not achieve a better than 2-approximation, and there slow algorithms with \O(\sqrt{m}) update time that achieve a better-than-2 approximation. Thus the obvious question is whether we can design an algorithm -- deterministic or randomized -- that achieves a tradeoff between these two: a o(m)o(\sqrt{m}) approximation and a better-than-2 approximation simultaneously. We answer this question in the affirmative for bipartite graphs. Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give stronger results for graphs whose arboricity is at most \al, achieving a (1+ \eps) approximation in worst-case time O(\al (\al + \log n)) for constant \eps. When the arboricity is constant, this bound is O(logn)O(\log n) and when the arboricity is polylogarithmic the update time is also polylogarithmic. The most important technical developement is the use of an intermediate graph we call an edge degree constrained subgraph (EDCS). This graph places constraints on the sum of the degrees of the endpoints of each edge: upper bounds for matched edges and lower bounds for unmatched edges. The main technical content of our paper involves showing both how to maintain an EDCS dynamically and that and EDCS always contains a sufficiently large matching. We also make use of graph orientations to help bound the amount of work done during each update.Comment: Longer version of paper that appears in ICALP 201

    Distributed PCP Theorems for Hardness of Approximation in P

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    We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment x{0,1}nx \in \{0,1\}^n to a CNF formula φ\varphi is shared between two parties, where Alice knows x1,,xn/2x_1, \dots, x_{n/2}, Bob knows xn/2+1,,xnx_{n/2+1},\dots,x_n, and both parties know φ\varphi. The goal is to have Alice and Bob jointly write a PCP that xx satisfies φ\varphi, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of xx. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of 2(logn)1o(1)2^{(\log n)^{1-o(1)}}; only (1+o(1))(1+o(1))-factor lower bounds (under SETH) were known before

    Optimal Dynamic Distributed MIS

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    Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees. The result trivially applies for the dynamic distributed model, in which edges or nodes may be inserted or deleted. In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either \emph{synchronous} or \emph{asynchronous}, with only \emph{a single adjustment} and in a single round, in expectation. These strong guarantees hold for the \emph{complete fully dynamic} setting: Insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former. Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy \emph{sequential} MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a 33-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing. Finally, our algorithm enjoys a useful \emph{history-independence} property, meaning the output is independent of the history of topology changes that constructed that graph. This means the output cannot be chosen, or even biased, by the adversary in case its goal is to prevent us from optimizing some objective function.Comment: 19 pages including appendix and reference

    On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

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    We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a one-pass deterministic online algorithm is 1/21/2, which is achieved by any greedy algorithm. D\"urr et al. recently presented a 22-pass algorithm called Category-Advice that achieves approximation ratio 3/53/5. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the kk-pass Category-Advice algorithm for all k1k \ge 1, and show that the approximation ratio converges to the inverse of the golden ratio 2/(1+5)0.6182/(1+\sqrt{5}) \approx 0.618 as kk goes to infinity. The convergence is extremely fast --- the 55-pass Category-Advice algorithm is already within 0.01%0.01\% of the inverse of the golden ratio. We then consider a natural greedy algorithm in the online stochastic IID model---MinDegree. This algorithm is an online version of a well-known and extensively studied offline algorithm MinGreedy. We show that MinDegree cannot achieve an approximation ratio better than 11/e1-1/e, which is guaranteed by any consistent greedy algorithm in the known IID model. Finally, following the work in Besser and Poloczek, we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model
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