Finding a Shortest Non-Zero Path in Group-Labeled Graphs

We study a constrained shortest path problem in group-labeled graphs with nonnegative edge length, called the non-zero problem. Depending on group question, this includes two types of tractable variants undirected graphs: one is parity-constrained path/cycle problem, and other computing noncontractible cycle surface-embedded graphs. For respect to finite abelian groups, Kobayashi Toyooka (2017) proposed randomized, pseudopolynomial-time algorithm via permanent computation. slightly more general class Yamaguchi (2016) showed reduction weighted linear matroid parity In particular, some cases are solved strongly polynomial time aid deterministic, polynomial-time for developed by Iwata (2021), which generalizes well-known fact that matching. paper, as first solution independent group, we present rather simple, This result captures common feature behind topological constraints The based Dijkstra’s unconstrained Edmonds’ blossom shrinking technique matching algorithms; approach inspired Derigs’ faster (1985) Furthermore, improve our so it does not require explicit shrinking, make computational match one. speeding-up step, dual programming formulation equivalent potential maximization plays key role.