Algorithms and Complexity for the Almost Equal Maximum Flow Problem

TL;DR

This paper investigates the computational complexity of the Almost Equal Maximum Flow Problem, revealing NP-completeness results and providing algorithms for specific cases, contrasting with the polynomial solvability of the equal maximum flow problem.

Contribution

It establishes NP-completeness of integer and fractional AEMFP, and offers polynomial algorithms for constant deviation and concave cases with fixed sets.

Findings

Integer AEMFP is NP-complete.

Fractional AEMFP with convex deviation is NP-complete.

Polynomial algorithms exist for constant deviation and concave cases with fixed sets.

Abstract

In the Equal Maximum Flow Problem (EMFP), we aim for a maximum flow where we require the same flow value on all edges in some given subsets of the edge set. In this paper, we study the closely related Almost Equal Maximum Flow Problems (AEMFP) where the flow values on edges of one homologous edge set differ at most by the valuation of a so called deviation function~$\Delta$. We prove that the integer almost equal maximum flow problem (integer AEMFP) is in general $\mathcal{NP}$-complete, and show that even the problem of finding a fractional maximum flow in the case of convex deviation functions is also $\mathcal{NP}$-complete. This is in contrast to the EMFP, which is polynomial time solvable in the fractional case. We provide inapproximability results for the integral AEMFP. For the integer AEMFP we state a polynomial algorithm for the constant deviation and concave case for a fixed number of homologous sets.