Fast algorithms for maximum subset matching and all-pairs shortest paths in graphs with a (not so) small vertex cover

Noga Alon, Raphael Yustcr

Research output: Chapter in Book/Report/Conference proceedingConference contribution

9 Scopus citations

Abstract

In the Maximum Subset Matching problem, which generalizes the maximum matching problem, we are given a graph G = (V, E) and S ⊂ V. The goal is to determine the maximum number of vertices of S that can be matched in a matching of G. Our first result is a new randomized algorithm for the Maximum Subset Matching problem that improves upon the fastest known algorithms for this problem. Our algorithm runs in Ṏ(ms(ω-1)/2) time if m > s(ω+1)/2 and in Õ(sω) time if m < s(ω+1)/2, where ω < 2.376 is the matrix multiplication exponent, m is the number of edges from S to V \ S, and s = |S|. The algorithm is based, in part, on a method for computing the rank of sparse rectangular integer matrices. Our second result is a new algorithm for the All-Pairs Shortest Paths (APSP) problem. Given an undirected graph with n vertices, and with integer weights from (1,...,W) assigned to its edges, we present an algorithm that solves the APSP problem in Õ(Wn ω(1,1,μ)) time where nμ = vc(G) is the vertex cover number of G and ω(1, 1, μ) is the time needed to compute the Boolean product of an n × n matrix with an n × nμ matrix. Already for the unweighted case this improves upon the previous O(n 2+μ) and Õ(nω) time algorithms for this problem. In particular, if a graph has a vertex cover of size O(n 0.29) then APSP in unweighted graphs can be solved in asymptotically optimal Õ(n2) time, and otherwise it can be solved in O(n 1.844vc(G)0.533) time. The common feature of both results is their use of algorithms developed in recent years for fast (sparse) rectangular matrix multiplication.

Original languageEnglish (US)
Title of host publicationAlgorithms - ESA 2007 - 15th Annual European Symposium, Proceedings
PublisherSpringer Verlag
Pages175-186
Number of pages12
ISBN (Print)9783540755197
DOIs
StatePublished - 2007
Event15th Annual European Symposium on Algorithms, ESA 2007 - Eilat, Israel
Duration: Oct 8 2007Oct 10 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4698 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other15th Annual European Symposium on Algorithms, ESA 2007
CountryIsrael
CityEilat
Period10/8/0710/10/07

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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