A Strongly Polynomial Algorithm for Finding a Shortest Non-zero Path in Group-Labeled Graphs
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Finding a shortest path or cycle in graphs is a fundamental problem. In this paper, we study the problem of finding a shortest non-zero path/cycle in group-labeled graphs with nonnegative edge length, which includes the following two types of tractable variants in undirected graphs, depending on the group in question. One is the parity-constrained shortest path/cycle problem, and the other is computing a shortest noncontractible cycle in surface-embedded graphs. For the shortest non-zero path problem with respect to finite abelian groups, Kobayashi and Toyooka (2017) proposed a randomized, pseudopolynomial algorithm via permanent computation. For a slightly more general class of groups, Yamaguchi (2016) showed a reduction of the problem to the weighted linear matroid parity problem. In particular, some cases are solved in strongly polynomial time via the reduction with the aid of a deterministic, polynomial algorithm for the weighted linear matroid parity problem developed by Iwata and Kobayashi (2017), which generalizes a well-known fact that the parity-constrained shortest path problem is solved via weighted matching. In this paper, as the first general solution independent of the group, we present a rather simple, deterministic, and strongly polynomial algorithm for the shortest non-zero path problem. The algorithm is based on Dijkstra's algorithm for the unconstrained shortest path problem and Edmonds' blossom shrinking technique in matching algorithms, and clarifies a common tractable feature behind the parity and topological constraints in the shortest path/cycle problem.
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