Edge-Minimum Walk of Modular Length in Polynomial Time

TL;DR

This paper introduces a polynomial-time algorithm for finding edge-minimum st-walks of a fixed modular length in directed graphs, and extends the approach to a generalized Steiner Network problem with modularity constraints.

Contribution

It is the first to study the modular length walk problem and provides a polynomial-time solution for fixed parameters, also extending to a generalized Steiner Network problem.

Findings

Polynomial-time algorithm for fixed r and q

Extension to modularity-constrained Steiner Network

First study of edge-minimum walks with modular length constraints

Abstract

We study the problem of finding, in a directed graph, an st-walk of length r mod q which is edge-minimum, i.e., uses the smallest number of distinct edges. Despite the vast literature on paths and cycles with modularity constraints, to the best of our knowledge we are the first to study this problem. Our main result is a polynomial-time algorithm that solves this task when r and q are constants. We also show how our proof technique gives an algorithm to solve a generalization of the well-known Directed Steiner Network problem, in which connections between endpoint pairs are required to satisfy modularity constraints on their length. Our algorithm is polynomial when the number of endpoint pairs and the modularity constraints on the pairs are constants.