Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Faster scaling algorithms for general graph matching problems
Journal of the ACM (JACM)
On the all-pairs-shortest-path problem in unweighted undirected graphs
Journal of Computer and System Sciences - Special issue on selected papers presented at the 24th annual ACM symposium on the theory of computing (STOC '92)
All pairs shortest paths for graphs with small integer length edges
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
Rectangular matrix multiplication revisited
Journal of Complexity
Fast rectangular matrix multiplication and applications
Journal of Complexity
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
A New Approach to Maximum Matching in General Graphs
ICALP '90 Proceedings of the 17th International Colloquium on Automata, Languages and Programming
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
All Pairs Shortest Paths in Undirected Graphs with Integer Weights
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Maximum Matchings via Gaussian Elimination
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Fast sparse matrix multiplication
ACM Transactions on Algorithms (TALG)
Colored intersection searching via sparse rectangular matrix multiplication
Proceedings of the twenty-second annual symposium on Computational geometry
Algebraic Structures and Algorithms for Matching and Matroid Problems
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
An O(v|v| c |E|) algoithm for finding maximum matching in general graphs
SFCS '80 Proceedings of the 21st Annual Symposium on Foundations of Computer Science
Approximating edge dominating set in dense graphs
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Approximating edge dominating set in dense graphs
Theoretical Computer Science
Fast matrix rank algorithms and applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Fast matrix rank algorithms and applications
Journal of the ACM (JACM)
Equistable simplicial, very well-covered, and line graphs
Discrete Applied Mathematics
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In the Maximum Subset Matching problem, which generalizesthe maximum matching problem, we are given a graph G = (V,E)and S ⊂ V. The goal is to determine the maximum number of verticesof S that can be matched in a matching of G. Our first result is a newrandomized algorithm for the Maximum Subset Matching problem thatimproves upon the fastest known algorithms for this problem. Our algorithmruns in Õ(ms(ω-1)/2) time if m ≥ s(ω+1)/2) and in Õ(sω) time ifm ≤ s(ω+1)/2), where ω mis the number of edges from S to V \ S, and s = |S|. The algorithm isbased, in part, on a method for computing the rank of sparse rectangularinteger matrices. Our second result is a new algorithm for the All-Pairs Shortest Paths(APSP) problem. Given an undirected graph with n vertices, and withinteger weights from {1,..., W } assigned to its edges, we present analgorithm that solves the APSP problem in Õ(Wnω(1,1,µ)) time wherenµ = υc(G) is the vertex cover number of G and ω(1, 1, µ) is the timeneeded to compute the Boolean product of an n×n matrix with an n×nµmatrix. Already for the unweighted case this improves upon the previousO(n2+µ) and Õ(nω) time algorithms for this problem. In particular, if agraph has a vertex cover of size O(n0.29) then APSP in unweighted graphscan be solved in asymptotically optimal Õ(n2) time, and otherwise it canbe solved in O(n1.844υc(G)0.533) time. The common feature of both results is their use of algorithms developedin recent years for fast (sparse) rectangular matrix multiplication.