Min-cost-flow preserving bijection between subgraphs and orientations

TL;DR

This paper presents a computationally efficient bijection between subgraphs and orientations of a graph that support a given flow, preserving costs and providing a new proof for a known combinatorial fact.

Contribution

It introduces an explicit, cost-preserving bijection between subgraphs and orientations supporting a specified flow, enhancing understanding of flow structures in graphs.

Findings

Established a bijection between subgraphs and orientations with $d$-flows.

The bijection preserves minimum-cost flows.

Provided an efficient, bijective proof of a known combinatorial identity.

Abstract

Consider an undirected graph $G=(V,E)$. A subgraph of $G$ is a subset of its edges, whilst an orientation of $G$ is an assignment of a direction to each edge. Provided with an integer circulation-demand $d:V\to \mathbb{Z}$, we show an explicit and efficiently computable bijection between subgraphs of $G$ on which a $d$-flow exists and orientations on which a $d$-flow exists. Moreover, given a cost function $w:E\to (0,\infty)$ we can find such a bijection which preserves the $w$-min-cost-flow. In 2013, Kozma and Moran showed, using dimensional methods, that the number of subgraphs $k$-connecting a vertex $s$ to a vertex $t$ is the same as the number of orientations $k$-connecting $s$ to $t$. An application of our result is an efficient, bijective proof of this fact.