Abstract-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
Furthermore, it was shown by Roditty and Vassilevska Williams REF that the problem (also for directed graphs) reduces to finding a minimum weight triangle in an undirected graph with Opnq nodes and weights in t1, . . . , OpM qu.
Minimum Weight Cycles and Triangles: Equivalences and Algorithms
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Mathematics Computer Science