Minimum Weight Cycles and Triangles: Equivalences and Algorithms

We consider the fundamental algorithmic problem of finding a cycle of minimum weight in a weighted graph. In particular, we show that the minimum weight cycle problem in an undirected n-node graph with edge weights in {1,...,M} or in a directed n-node graph with edge weights in {-M,..., M} and no negative cycles can be efficiently reduced to finding a minimum weight _triangle_ in an Theta(n)-node _undirected_ graph with weights in {1,...,O(M)}. Roughly speaking, our reductions imply the following surprising phenomenon: a minimum cycle with an arbitrary number of weighted edges can be ``encoded'' using only three edges within roughly the same weight interval! This resolves a longstanding open problem posed in a seminal work by Itai and Rodeh [SIAM J. Computing 1978] on minimum cycle in unweighted graphs. A direct consequence of our efficient reductions are tilde{O}(Mn^{\omega})0) for minimum weight cycle immediately implies a O(n^{3-\delta})-time algorithm (\delta>0) for APSP.