Minimum-cost integer circulations in given homology classes

TL;DR

This paper studies the problem of finding minimum-cost integer circulations in a directed graph embedded on a surface, providing complexity results and algorithms for orientable and non-orientable surfaces.

Contribution

It offers a polyhedral description for orientable surfaces, proves NP-hardness for non-orientable surfaces, and presents a polynomial-time algorithm for fixed-genus non-orientable surfaces.

Findings

Polyhedral description of feasible solutions for orientable surfaces.

NP-hardness of the problem on non-orientable surfaces.

Polynomial-time algorithm for fixed-genus non-orientable surfaces.

Abstract

Let $D$ be a directed graph cellularly embedded in a surface together with non-negative cost on its arcs. Given any integer circulation in $D$, we study the problem of finding a minimum-cost non-negative integer circulation in $D$ that is homologous over the integers to the given circulation. A special case of this problem arises in recent work on the stable set problem for graphs with bounded odd cycle packing number, in which the surface is non-orientable (Conforti et al., SODA'20). For orientable surfaces, polynomial-time algorithms have been obtained for different variants of this problem. We complement these results by showing that the convex hull of feasible solutions has a very simple polyhedral description. In contrast, only little seems to be known about the case of non-orientable surfaces. We show that the problem is strongly NP-hard for general non-orientable surfaces, and give the first polynomial-time algorithm for surfaces of fixed genus. For the latter, we provide a characterization of homology (over the integers) that allows us to recast the problem as a special integer program, which can be efficiently solved using proximity results and dynamic programming.