On r-Simple k-Path

Abstract

An r-simple k-path is a path in the graph of length k that passes through each vertex at most r times. The r-SIMPLE k-PATH problem, given a graph G as input, asks whether there exists an r-simple k-path in G. We first show that this problem is NP-Complete. We then show that there is a graph G that contains an r-simple k-path and no simple path of length greater than 4logk/logr. So this, in a sense, motivates this problem especially when one’s goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex.

We then give a randomized algorithm that runs in time

$$\mathrm{poly}(n)\cdot 2^{O( k\cdot \log r/r)}$$

that solves the r-SIMPLE k-PATH on a graph with n vertices with one-sided error. We also show that a randomized algorithm with running time poly(n)·2^{(c/2)k/ r} with c < 1 gives a randomized algorithm with running time poly(n)·2^cn for the Hamiltonian path problem in a directed graph - an outstanding open problem. So in a sense our algorithm is optimal up to an O(logr) factor in the exponent.

The crux of our method is to use low degree testing to efficiently test whether a polynomial contains a monomial where all individual degrees are bounded by a given r.