Arc connectivity and submodular flows in digraphs

Abdi, Ahmad ORCID: 0000-0002-3008-4167, Cornuéjols, Gérard and Zambelli, Giacomo (2024) Arc connectivity and submodular flows in digraphs. Combinatorica. ISSN 0209-9683

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Abstract

Let D=(V,A) be a digraph. For an integer k≥1, a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer τ≥1, suppose d A +(U)+(τk-1)d A -(U)≥τ for all U⊊V,U≠∅, where d A +(U) and d A -(U) denote the number of arcs in A leaving and entering U, respectively. Let C be a crossing family over ground set V, and let f:C→Z be a crossing submodular function such that f(U)≥kτ(d A +(U)-d A -(U)) for all U∈C. Then D has a k-arc-connected flip J such that f(U)≥d J +(U)-d J -(U) for all U∈C. The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is τ-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

Item Type:	Article
Official URL:	https://link.springer.com/journal/493
Additional Information:	© 2024 The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	01 May 2024 15:09
Last Modified:	16 Jul 2024 00:06
URI:	http://eprints.lse.ac.uk/id/eprint/122866

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