Min-cost-flow preserving bijection between subgraphs and orientations

12/16/2021

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by   Izhak Elmaleh, et al.

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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→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→ (0,∞) 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.