TY - GEN

T1 - Fast algorithms for maximum subset matching and all-pairs shortest paths in graphs with a (not so) small vertex cover

AU - Alon, Noga

AU - Yustcr, Raphael

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38049057332&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38049057332&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-75520-3_17

DO - 10.1007/978-3-540-75520-3_17

M3 - Conference contribution

AN - SCOPUS:38049057332

SN - 9783540755197

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 175

EP - 186

BT - Algorithms - ESA 2007 - 15th Annual European Symposium, Proceedings

PB - Springer Verlag

T2 - 15th Annual European Symposium on Algorithms, ESA 2007

Y2 - 8 October 2007 through 10 October 2007

ER -