Journal of the ACM (JACM)
Finding Even Cycles Even Faster
SIAM Journal on Discrete Mathematics
Fast Estimation of Diameter and Shortest Paths (Without Matrix Multiplication)
SIAM Journal on Computing
All-Pairs Almost Shortest Paths
SIAM Journal on Computing
Journal of Algorithms
Detecting short directed cycles using rectangular matrix multiplication and dynamic programming
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Efficient approximation algorithms for shortest cycles in undirected graphs
Information Processing Letters
Subcubic Equivalences between Path, Matrix and Triangle Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
On Dynamic Shortest Paths Problems
Algorithmica
All-pairs nearly 2-approximate shortest-paths in O(n2 polylog n) time
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
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This article considers the problem of computing a minimum weight cycle in weighted undirected graphs. Given a weighted undirected graph G = (V,E,w), let C be a minimum weight cycle of G, let w(C) be the weight of C, and let wmax(C) be the weight of the maximum edge of C. We obtain three new approximation algorithms for the minimum weight cycle problem: (1) for integral weights from the range [1,M], an algorithm that reports a cycle of weight at most 4 3w(C) in O(n2 log n(log n + log M)) time; (2) For integral weights from the range [1,M], an algorithm that reports a cycle of weight at most w(C) + wmax(C) in O(n2 log n(log n + log M)) time; (3) For nonnegative real edge weights, an algorithm that for any ε 0 reports a cycle of weight at most (4 3 + ε)w(C) in O(1 ε n2 log n(log log n)) time. In a recent breakthrough, Williams and Williams [2010] showed that a subcubic algorithm, that computes the exact minimum weight cycle in undirected graphs with integral weights from the range [1,M], implies a subcubic algorithm for computing all-pairs shortest paths in directed graphs with integral weights from the range [−M,M]. This implies that in order to get a subcubic algorithm for computing a minimum weight cycle, we have to relax the problem and to consider an approximated solution. Lingas and Lundell [2009] were the first to consider approximation in the context of minimum weight cycle in weighted graphs. They presented a 2-approximation algorithm for integral weights with O(n2 log n(log n + log M)) running time. They also posed, as an open problem, the question whether it is possible to obtain a subcubic algorithm with a c-approximation, where c ε)-approximation, where 0 ε ≤ 1/3, implies a subcubic combinatorial algorithm for multiplying two boolean matrices.