A Complete Characterization of Convexity in Flow Games

TL;DR

This paper provides a complete characterization of convexity in flow games, linking structural network conditions to convexity and enabling polynomial-time recognition.

Contribution

It introduces necessary and sufficient structural conditions for flow game convexity and proves their equivalence to dual separability.

Findings

Flow game convexity characterized by acyclicity, bottleneck exclusivity, and capacity sufficiency.

Convexity recognition can be performed in polynomial time.

Structural conditions resolve the paradox between cycle orientations and convexity.

Abstract

Flow games coincide precisely with the fundamental class of non-negative totally balanced games. However, the conditions for their convexity have remained elusive. In this paper, we resolve this challenge by providing a complete characterization. Specifically, we show that a flow game is convex if and only if its underlying network satisfies three structural conditions: acyclicity, bottleneck exclusivity, and capacity sufficiency. These structural conditions are also equivalent to dual separability, which resolves the apparent paradox between cycle orientations and game-theoretic convexity by decoupling path contributions via bottleneck exclusivity. Furthermore, our characterization yields an efficient recognition procedure, establishing that flow game convexity is verifiable in polynomial time.