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On approximating the longest path in a graph. (English) Zbl 0876.68083
Summary: We consider the problem of approximating the longest path in undirected graphs. In an attempt to pin down the best achievable performance ratio of an approximation algorithm for this problem, we present both positive and negative results. First, a simple greedy algorithm is shown to find long paths in dense graphs. We then consider the problem of finding paths in graphs that are guaranteed to have extremely long paths. We devise an algorithm that finds paths of a logarithmic length in Hamiltonian graphs. This algorithm works for a much larger class of graphs (weakly Hamiltonian), where the result is the best possible. Since the hard case appears to be that of sparse graphs, we also consider sparse random graphs. Here, we show that a relatively long path can be obtained, thereby partially answering an open problem of A. Broder, A. M. Frieze and E. Shamir [Finding hidden Hamiltonian cycles, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, 182-189 (1991)].
To explain the difficulty of obtaining better approximations, we also prove hardness results. We show that, for any \(\varepsilon<1\), the problem of finding a path of length \(n-n^\varepsilon\) in an \(n\)-vertex Hamiltonian graph is NP-hard. We then show that no polynomial-time algorithm can find a constant factor approximation to the longest-path problem unless \(\text{P}=\text{NP}\). We conjecture that the result can be strengthened to say that, for some constant \(\delta>0\), finding an approximation of ratio \(n^\delta\) is also NP-hard. As evidence toward this conjecture, we show that if any polynomial-time algorithm can approximate the longest path to a ratio of \(2^{O(\log^{1-\varepsilon}n)}\) for any \(\varepsilon>0\), then NP has a quasi-polynomial deterministic time simulation. The hardness results apply even to the special case where the input consists of bounded degree graphs.

MSC:
68R10 Graph theory (including graph drawing) in computer science
68W10 Parallel algorithms in computer science
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