On sparse spanners of weighted graphs
Discrete & Computational Geometry
Journal of the ACM (JACM)
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
Finding Even Cycles Even Faster
ICALP '94 Proceedings of the 21st International Colloquium on Automata, Languages and Programming
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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
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
A shortest cycle for each vertex of a graph
Information Processing Letters
Subquadratic time approximation algorithms for the girth
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Optimal distributed all pairs shortest paths and applications
PODC '12 Proceedings of the 2012 ACM symposium on Principles of distributed computing
Distributed algorithms for network diameter and girth
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
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This paper 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 maximal 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 non-negative 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 Vassilevska Williams and Williams [WW10] 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 [LL09] 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 Surprisingly, the approximation factor of 4/3 is not accidental. We show using the new result of Vassilevska Williams and Williams [WW10] that a subcubic combinatorial algorithm with (4/3 − ε)-approximation, where 0