TY - GEN

T1 - A strongly polynomial algorithm for finding a shortest non-zero path in group-labeled graphs

AU - Yamaguchi, Yutaro

PY - 2020

Y1 - 2020

N2 - We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. Depending on the group in question, this problem includes two types of tractable variants in undirected graphs: 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.

AB - We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the shortest non-zero path problem. Depending on the group in question, this problem includes two types of tractable variants in undirected graphs: 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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M3 - Conference contribution

AN - SCOPUS:85084054297

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1923

EP - 1932

BT - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

A2 - Chawla, Shuchi

PB - Association for Computing Machinery

T2 - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

Y2 - 5 January 2020 through 8 January 2020

ER -