How hard is it to marry at random? (On the approximation of the permanent)
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
An algorithm for finding Hamilton cycles in random graphs
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Conductance and the rapid mixing property for Markov chains: the approximation of permanent resolved
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Almost-optimum speed-ups of algorithms for bipartite matching and related problems
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Probabilistic analysis of network flow algorithms
Mathematics of Operations Research
Probabilistic analysis of matching and network flow algorithms
Probabilistic analysis of matching and network flow algorithms
Fast parallel matching in expander graphs
SPAA '93 Proceedings of the fifth annual ACM symposium on Parallel algorithms and architectures
Graph partitioning using single commodity flows
Journal of the ACM (JACM)
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Hall's Theorem states that a bipartite graph has a perfect matching if and only if every set of vertices has an equal number of neighbours. Equivalently, it states that every non-maximum matching has an augmenting path if the graph is an expander with expansion 1. We use this insight to demonstrate that if a graph is an expander with expansion more than one than every non-maximum matching has a short augmenting path and, therefore, the bipartite matching algorithm performs much better on such graphs than in the worst case. We then apply this idea to the average case analysis of various augmenting path algorithms and to the approximation of the permanent. In particular, we demonstrate that the following algorithms perform much better on the average than in the worst case. In fact, they will rarely exhibit their worst-case running times.Hopcroft-Karp's algorithm for bipartite matchings.Micali-Vazirani's and Even-Kariv's algorithms for non-bipartite matchings.Gabow-Tarjan's parallel algorithm for bipartite matchings.Dinic's algorithm for k-factors and 0-1 network flows.Jerrum-Sinclair's approximation scheme for the permanent.It seems rather surprising that the algorithms which are the fastest known for worst-case inputs also do exceedingly on almost every graph.