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 ε<1, the problem of finding a path of length n−n ^ε in an n-vertex Hamiltonian graph is shown to be NP-hard. We also show that no polynomial time algorithm can find a constant factor approximation to the longest path problem unless P=NP. Finally, it is shown that if any polynomial time algorithm can approximate the longest path to a ratio of 2^O(log1−ε n), for any ε>0, then NP has a quasi-polynomial deterministic time simulation.

Supported by NSF Grant CCR-9010517, and grants from Mitsubishi and OTL.

Supported an NSF Graduate Fellowship.

Supported by a grant from Toshiba Corporation.