# Finding a Shortest Non-zero Path in Group-Labeled Graphs

## TL;DR

This paper introduces a simple, deterministic, and strongly polynomial-time algorithm for finding the shortest non-zero path in group-labeled graphs, extending previous work by leveraging Dijkstra's algorithm and blossom shrinking techniques.

## Contribution

It provides the first general, group-independent, strongly polynomial-time algorithm for the shortest non-zero path problem, improving computational efficiency and simplicity.

## Key findings

- The algorithm is based on Dijkstra's shortest path and Edmonds' blossom shrinking.
- It does not require explicit blossom shrinking, matching Derigs' algorithm speed.
- A dual linear programming approach enhances the algorithm's speed.

## 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. 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.