Finding a Shortest Non-zero Path in Group-Labeled Graphs via Permanent Computation

Abstract

A group-labeled graph is a directed graph with each arc labeled by a group element, and the label of a path is defined as the sum of the labels of the traversed arcs. In this paper, we propose a polynomial time randomized algorithm for the problem of finding a shortest s-t path with a non-zero label in a given group-labeled graph (which we call the Shortest Non-Zero Path Problem). This problem generalizes the problem of finding a shortest path with an odd number of edges, which is known to be solvable in polynomial time by using matching algorithms. Our algorithm for the Shortest Non-Zero Path Problem is based on the ideas of Bj?rklund and Husfeldt (Proceedings of the 41st international colloquium on automata, languages and programming, part I. LNCS 8572, pp 211---222, 2014). We reduce the problem to the computation of the permanent of a polynomial matrix modulo two. Furthermore, by devising an algorithm for computing the permanent of a polynomial matrix modulo $$2^r$$2r for any fixed integer r, we extend our result to the problem of packing internally-disjoint s-t paths.