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 Θ(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 Õ(Mn^ω) ≤ 6(Mn^2.376)-time algorithms using fast matrix multiplication (FMM) for finding a minimum weight cycle in both undirected graphs with integral weights from the interval [1, M] and directed graphs with integral weights from the interval [-M,M]. The latter seems to reveal a strong separation between the all pairs shortest paths (APSP) problem and the minimum weight cycle problem in directed graphs as the fastest known APSP algorithm has a running time of O(M ^0.681n^2.575) by Zwick [J. ACM 2002]. In contrast, when only combinatorial algorithms are allowed (that is, without FMM) the only known solution to minimum weight cycle is by computing APSP. Interestingly, any separation between the two problems in this case would be an amazing breakthrough as by a recent paper by Vassilevska W. and Williams [FOCS'10], any O(n3^-ε)-time algorithm (ε >; 0) for minimum weight cycle immediately implies a O(n^3-δ)-time algorithm (δ >; 0) for APSP.

1. Introduction

We consider the algorithmic problem of finding a cycle of minimum weight in weighted directed and undirected graphs. Surprisingly, although the problem is very fundamental, the state of the art for it dates back to a seminal paper by Itai and Rodeh [10] from the 1970s, that deals only with the unweighted variant of the problem. Itai and Rodeh presented an -time algorithm for an n-node unweighted undirected graph and an -time algorithm for an n-node unweighted directed graph. (Here is the exponent of square matrix multiplication over a ring; [4].) In the same paper, Itai and Rodeh posed the question whether similar results exist for weighted graphs. In this paper we provide a positive answer to this longstanding open problem by presenting -time algorithms for directed graphs with integral edge weights in (and no negative cycles) and for undirected graphs with integral edge weights in .