A Dijkstra-Based Efficient Algorithm for Finding a Shortest Non-zero Path in Group-Labeled Graphs
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The parity constrained shortest path problem is a well-known variant that is tractable via weighted matching if a graph is undirected and has nonnegative edge length. As a generalization, we focus on the problem of finding a shortest non-zero path between two specified vertices s and t in a given group-labeled graph. For the case when the group Γ in question is finite and abelian, Kobayashi and Toyooka (2017) proposed a randomized, pseudopolynomial-time algorithm via permanent computation. For a slightly more general case when Γ is finitely generated and abelian, 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-time algorithm for the weighted linear matroid parity problem due to Iwata and Kobayashi (2017), which generalizes the aforementioned fact that the parity constrained shortest path problem in undirected graphs is solved via weighted matching. In this paper, as the first general solution independently of the group Γ, we present a rather simple, deterministic, and strongly polynomial-time algorithm for the shortest non-zero path problem in group-labeled graphs. The basic strategy is as follows: we first compute a shortest path tree rooted at s, and if the unique s--t path in the tree is not non-zero, then we find a shortest s--t path among those who have different labels. The former part is done just by Dijkstra's algorithm, and the key technique in the latter part is like Edmonds' blossom shrinking algorithm for matching problems.
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