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.

我们研究了具有非负边长的组标记图中的约束最短路径问题，称为最短非零路径问题。根据所讨论的组，该问题包括无向图中的两种类型的易处理变体：一种是奇偶约束最短路径/循环问题，另一种是计算曲面嵌入图中的最短不可收缩循环。

对于关于有限阿贝尔群的最短非零路径问题，Kobayashi 和 Toyooka (2017) 通过永久计算提出了一种随机伪多项式时间算法。对于更一般的组，Yamaguchi (2016) 将问题简化为加权线性拟阵奇偶校验问题。特别是，Iwata 和 Kobayashi (2021) 开发的加权线性拟阵奇偶校验问题的确定性多项式时间算法通过约简在强多项式时间内解决了某些情况，该算法推广了一个众所周知的事实，即奇偶约束最短路径问题通过加权匹配解决。

在本文中，作为第一个独立于群的通用解，我们针对最短非零路径问题提出了一种相当简单、确定性和强多项式时间的算法。该结果捕获了最短路径/周期问题中奇偶性和拓扑约束背后的常见易处理特征。该算法基于无约束最短路径问题的Dijkstra算法和匹配算法中的Edmonds开花收缩技术；这种方法受到 Derigs 的更快算法 (1985) 的启发，该算法通过减少加权匹配来解决奇偶约束最短路径问题。此外，我们改进了我们的算法，使其不需要显式的开花收缩，并使计算时间与 Derigs 的计算时间相匹配。在加速步骤中，