Lattice Structure and Efficient Basis Construction for Strongly Connected Orientations

TL;DR

This paper presents a polynomial-time combinatorial algorithm to construct an integral basis for tight strongly connected orientations in bidirected graphs, extending prior non-constructive results and enabling efficient solutions for related parity-constrained problems.

Contribution

It provides the first constructive, polynomial-time method to find an integral basis for tight SCOs, advancing the understanding of their linear structure.

Findings

Polynomial-time algorithm for basis construction

Extension of non-constructive existence proof to a constructive method

Application to parity-constrained tight SCO problems

Abstract

Let $\vec{G}=(V,E^+\cup E^-)$ be a bidirected graph whose underlying undirected graph $G=(V,E)$ is $2$-edge-connected. A strongly connected orientation (SCO) is defined as a subset of arcs that contains exactly one of $e^+,e^-$ for every $e\in E$ and induces a strongly connected subgraph of $\vec{G}$. Given a family $\mathcal{F}$ of proper subsets of $V$, we call an SCO tight if there is exactly one arc entering $U$ for every $U\in \mathcal{F}$. We give a polynomial-time algorithm to construct a set $\mathcal{B}$ consisting of tight SCO's which forms an integral basis for the linear hull of tight SCO's. This means that $\mathcal{B}$ is a linearly independent subset of tight SCO's, and every integer vector in the linear hull of tight SCO's can be written as an integral combination of $\mathcal{B}$. This extends the main result of Abdi, Conu\'ejols, Liu and Silina (IPCO 2025), who gave a non-constructive proof of the existence of such a basis in an equivalent setting. While their proof uses polyhedral theory, our proof is purely combinatorial and yields a polynomial-time algorithm. As an application of our algorithm, we show that parity-constrained tight strongly connected orientation can be solved in deterministic polynomial time. Along the way, we discover appealing connections to the theory of perfect matching lattices.