Lattice Structure and Efficient Basis Construction for Strongly Connected Orientations
This work addresses a combinatorial optimization problem in graph theory, offering algorithmic improvements for researchers in discrete mathematics and operations research, though it is incremental relative to existing theoretical foundations.
The paper tackles the problem of constructing an integral basis for tight strongly connected orientations (SCOs) in bidirected graphs, providing a polynomial-time algorithm that extends prior non-constructive results and enables deterministic polynomial-time solutions for parity-constrained tight SCOs.
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, Conuéjols, 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.