On approximating the longest path in a graph

Abstract

We consider the problem of approximating the longest path in undirected graphs and present both positive and negative results. A simple greedy algorithm is shown to find long paths in dense graphs. We also present an algorithm for finding paths of a logarithmic length in weakly Hamiltonian graphs, and this result is the best possible. For sparse random graphs, we show that a relatively long path can be obtained. To explain the difficulty of obtaining better approximations, we provide some strong hardness results. For any e 0, then NP has a quasi-polynomial deterministic time simulation.