Strongly connected orientations and integer lattices

TL;DR

This paper explores the lattice properties of certain polytopes related to strongly connected orientations in digraphs, revealing new structural insights and implications for hypergraph orientations and a conjecture by Woodall.

Contribution

It establishes a surprising lattice basis property for faces of a polytope associated with strongly connected orientations, and develops a matching theory-like framework for degree-constrained dijoins.

Findings

The face $F$ contains an integral basis under mild conditions.

A relaxation of Woodall's conjecture is proved using $p$-adic packings.

The all-ones vector is in the lattice generated by $F$.

Abstract

Let $D=(V,A)$ be a digraph whose underlying undirected 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 $a\in A,i\in \{0,1\}$. We prove under a mild necessary condition that $F\cap \{0,1\}^A$ contains an integral basis $B$, 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 $\tau$, is equal to the maximum number of disjoint dijoins. We prove a relaxation of this conjecture, by finding for any prime number $p\geq 2$, a $p$-adic packing of dijoins of value $\tau$ and of support size at most $2|A|$. We also prove that the all-ones vector belongs to the lattice generated by $F\cap \{0,1\}^A$, where $F$ is the face of $P$ satisfying $x(\delta^+(U))=1$ for every dicut $\delta^+(U)$ with minimum size.