Strongly connected orientations and integer lattices

Abdi, Ahmad ORCID: 0000-0002-3008-4167, Cornuéjols, Gérard, Liu, Siyue and Silina, Olha (2025) Strongly connected orientations and integer lattices. In: Megow, Nicole and Basu, Amitabh, (eds.) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science,15620 (15620). Springer. ISBN 9783031931116

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Abstract

Let D=(V,A) be a digraph whose underlying graph is 2-edge-connected, and let P be the polytope whose vertices are the incidence vectors of arc sets whose reversal makes D strongly connected. We study the lattice theoretic properties of the integer points contained in a proper face F of P not contained in {x:x a=i} for any a∈A,i∈{0,1}. We prove under a mild necessary condition that F∩{0,1} A contains an integral basisB, i.e., B is linearly independent, and any integral vector in the linear hull of F is an integral linear combination of B. This result is surprising as the integer points in F do not necessarily form a Hilbert basis. In proving the result, we develop a theory similar to Matching Theory for degree-constrained dijoins in bipartite digraphs. Our result has consequences for head-disjoint strong orientations in hypergraphs, and also to a famous conjecture by Woodall that the minimum size of a dicut of D, say τ, is equal to the maximum number of disjoint dijoins. We prove a relaxation of this conjecture, by finding for any prime number p≥2, a p-adic packing of dijoins of value τ and of support size at most 2|A|. We also prove that the all-ones vector belongs to the lattice generated by F∩{0,1} A, where F is the face of P satisfying x(δ +(U))=1 for every minimum dicut δ +(U).

Item Type:	Book Section
Additional Information:	© 2025 The Author
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	08 Apr 2025 14:24
Last Modified:	11 Jul 2025 18:24
URI:	http://eprints.lse.ac.uk/id/eprint/127860

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