Finding Long Paths, Cycles and Circuits

Abstract

We present a polynomial-time algorithm to find a cycle of length $\exp(\Omega(\sqrt{\log \ell}))$ in an undirected graph having a cycle of length ≥ ℓ. This is a slight improvement over previously known bounds. In addition the algorithm is more general, since it can similarly approximate the longest circuit, as well as the longest S-circuit (i.e., for S an arbitrary subset of vertices, a circuit that can visit each vertex in S at most once). We also show that any algorithm for approximating the longest cycle can approximate the longest circuit, with a square root reduction in length. For digraphs, we show that the long cycle and long circuit problems have the same approximation ratio up to a constant factor. We also give an algorithm to find a vw-path of length ≥ logn/loglogn if one exists; this is within a loglogn factor of a hardness result.