Min-cost-flow preserving bijection between subgraphs and orientations

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.