## Abstract

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-time 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-time algorithm for the weighted linear matroid parity problem developed by Iwata and Kobayashi (2021), 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-time algorithm for the shortest non-zero path problem. This result captures a common tractable feature behind the parity and topological constraints in the shortest path/cycle problem. The algorithm is based on Dijkstra’s algorithm for the unconstrained shortest path problem and Edmonds’ blossom shrinking technique in matching algorithms; this approach is inspired by Derigs’ faster algorithm (1985) for the parity-constrained shortest path problem via a reduction to weighted matching. Furthermore, we improve our algorithm so that it does not require explicit blossom shrinking, and make the computational time match Derigs’ one. In the speeding-up step, a dual linear programming formulation of the equivalent problem based on potential maximization for the unconstrained shortest path problem plays a key role.

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## Acknowledgments

The authors are deeply grateful to the anonymous reviewers of this paper and the preliminary version [25] for their valuable comments and suggestions. This work was partially supported by RIKEN Center for Advanced Intelligence Project.

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## Additional information

A preliminary version [25] of this paper appeared in SODA 2020.

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Iwata, Y., Yamaguchi, Y. Finding a Shortest Non-Zero Path in Group-Labeled Graphs.
*Combinatorica* **42**
(Suppl 2), 1253–1282 (2022). https://doi.org/10.1007/s00493-021-4736-x

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DOI: https://doi.org/10.1007/s00493-021-4736-x

### Mathematics Subject Classification (2010)

- 05C85
- 90C27
- 05C22
- 05C38
- 05E15
- 55P10