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
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
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 -