A Local Computation Approximation Scheme to Maximum Matching

Mansour, Yishay; Vardi, Shai

doi:10.1007/978-3-642-40328-6_19

Cited by 35 publications

(37 citation statements)

References 22 publications

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“…This random seed is a costly resource and we try to minimize its length. Known randomised implementations of greedy algorithms in the CentLOCAL domain require explicit random ordering constructions [3,30] over the vertices or edges [3,22,23,30]. In our implementation of the simulation, on the other hand, we use a permutation over the labels, which is a stronger requirement than a random ordering.…”

Section: Noveltymentioning

confidence: 99%

Non-local Probes Do Not Help with Many Graph Problems

Göös

Hirvonen

Levi³

et al. 2016

Lecture Notes in Computer Science

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This work bridges the gap between distributed and centralised models of computing in the context of sublinear-time graph algorithms. A priori, typical centralised models of computing (e.g., parallel decision trees or centralised local algorithms) seem to be much more powerful than distributed message-passing algorithms: centralised algorithms can directly probe any part of the input, while in distributed algorithms nodes can only communicate with their immediate neighbours. We show that for a large class of graph problems, this extra freedom does not help centralised algorithms at all: for example, efficient stateless deterministic centralised local algorithms can be simulated with efficient distributed message-passing algorithms. In particular, this enables us to transfer existing lower bound results from distributed algorithms to centralised local algorithms. arXiv:1512.05411v1 [cs.DS] 16 Dec 2015A lot of recent work on efficient graph algorithms for massive graphs can be broadly classified in one of the following categories:1. Probe-query models [1, 3, 7-9, 16, 22, 31]: Here typical applications are related to large-scale network analysis: we have a huge storage system in which the input graph is stored, and a computer that can access the storage system. The user of the computer can make queries related to the properties of the graph.Conceptually, we have two separate entities: the input graph and a computer. Initially, the computer is unaware of the structure of the graph, but it can probe it to learn more about the structure of the graph. Typically, the goal is to answer queries after a sublinear number of probes. 2.Message-passing models [6,14,15,19,21,26,29,34]: In message-passing models, typical applications are related to controlling large computer networks: we have a computer network (say, the Internet) that consists of a large number of network devices, and the devices need to collaborate to solve a graph problem related to the structure of the network so that each node knows its own part of the solution when the algorithm stops.Conceptually, each node of the input graph is a computational entity. Initially, the nodes are only aware of their own identity and the connections to their immediate neighbours, but the nodes can exchange messages with their neighbours in order to learn more about the structure of the graph. Typically, the goal is to solve graph problems in a sublinear number of communication rounds.

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Section: Noveltymentioning

confidence: 99%

Non-local Probes Do Not Help with Many Graph Problems

Göös

Hirvonen

Levi³

et al. 2016

Lecture Notes in Computer Science

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show abstract

“…Very efficient solutions are known, some of which yield constant factor approximation to maximum matching size in poly(d) time, where d is the maximum degree of the graph (see, e.g. [17,19,4,18,15] and references therein). None of these algorithms, however, seem directly amenable to the streaming model when the underlying graph has unbounded degree.…”

Section: Related Workmentioning

confidence: 99%

Approximating matching size from random streams

Kapralov¹,

Khanna²,

Sudan³

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present a streaming algorithm that makes one pass over the edges of an unweighted graph presented in random order, and produces a polylogarithmic approximation to the size of the maximum matching in the graph, while using only polylogarithmic space. Prior to this work the only approximations known were a folkloreÕ( √ n) approximation with polylogarithmic space in an n vertex graph and a constant approximation with Ω(n) space. Our work thus gives the first algorithm where both the space and approximation factors are smaller than any polynomial in n.Our algorithm is obtained by effecting a streaming implementation of a simple "local" algorithm that we design for this problem. The local algorithm produces a O(k · n 1/k ) approximation to the size of a maximum matching by exploring the radius k neighborhoods of vertices, for any parameter k. We show, somewhat surprisingly, that our local algorithm can be implemented in the streaming setting even for k = Ω(log n/ log log n). Our analysis exposes some of the problems that arise in such conversions of local algorithms into streaming ones, and gives techniques to overcome such problems.

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“…[MV13], showed that there exists a (1 − ǫ)-approximation algorithm to maximum matching. Unfortunately, that algorithm does not yield a monotonic allocation.…”

Section: Open Questionmentioning

confidence: 99%

Local computation mechanism design

Hassidim

Mansour

Vardi

2014

Proceedings of the Fifteenth ACM Conference on Economics and Computation

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We introduce the notion of local computation mechanism design -designing game theoretic mechanisms that run in polylogarithmic time and space. Local computation mechanisms reply to each query in polylogarithmic time and space, and the replies to different queries are consistent with the same global feasible solution. When the mechanism employs payments, the computation of the payments is also done in polylogarithmic time and space. Furthermore, the mechanism needs to maintain incentive compatibility with respect to the allocation and payments.We present local computation mechanisms for a variety of classical game-theoretical problems: (1) stable matching, (2) job scheduling, (3) combinatorial auctions for unit-demand and k-minded bidders, and (4) the housing allocation problem.For stable matching, some of our techniques may have implications to the global (non-LCA) setting. Specifically, we show that when the men's preference lists are bounded, we can achieve an arbitrarily good approximation to the stable matching within a fixed number of iterations of the Gale-Shapley algorithm.

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A Local Computation Approximation Scheme to Maximum Matching

Cited by 35 publications

References 22 publications

Non-local Probes Do Not Help with Many Graph Problems

Non-local Probes Do Not Help with Many Graph Problems

Approximating matching size from random streams

Local computation mechanism design

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