Integrality in the multinetwork min‐cost equal‐flow problem

• Published in: Networks Vol. 80; no. 3; pp. 267 - 273
• Main Authors: Hassin, Refael, Poznanski, Renata
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.10.2022; Wiley Subscription Services, Inc
• Subjects: computational complexity; Cost function; equal‐flow problem; Graph theory; Integers; integral solutions; min‐cost flow; Reproduction (copying); Series (mathematics)
• ISSN: 0028-3045; 1097-0037
• DOI: 10.1002/net.22094

We consider the min‐cost flow problem on multiple networks defined on duplicates of the same directed graph. The problem is to compute feasible flows such that the sum of flow costs is minimized subject to network demands and coupling constraints that force identical flows on duplicate copies of the same edge for a subset of “special” edges. We focus on whether such a problem has an integer optimal solution. For the case of integer capacities and demands and flow equality on a single edge, there is always an integer optimal solution and it can be found efficiently. For flow equality on multiple edges, we characterize the cases, with respect to the number of special edges and the number of networks, where if the graph is series‐parallel then an integer optimal solution exists for all possible capacity and cost functions.