Lattice Structure and Efficient Basis Construction for Strongly Connected Orientations

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, 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.

Figures (6)

• Figure 1: The solid boxes are types of digrafts. The dashed boxes are steps involved in finding a specific type of dicuts. The dotted boxes are steps executed to reduce the digrafts.
• Figure 2: (Left): an example of $2$-separation dicuts. The non-tight vertices (i.e.,$S\setminus S^t$) are filled solid; the vertices of a $2$-separation are circled; the two arising tight dicuts are in dashed. (Right): an example of barrier dicuts. The vertices of the barrier are circled and the arising barrier dicuts are in dashed.
• Figure 3: Illustration of a set $X$ found in \ref{['lemma:violate_robust']} and a set $Y_1$ violating the condition of \ref{['lemma:tight_edge_cover']}.
• Figure 4: $(D_1,S_1^t)$ with a set $X$ circled on the left side; dicuts $F_1,\ldots, F_k$ are in dashed.
• Figure 5: The contraction $(D_2,S_2^t)$ with a barrier $X$ circled on the left; dicuts $F_1,\ldots, F_k$ are in dashed. All of $F_i$ except one are trivial.

Theorems & Definitions (118)

• Theorem 1.1
• Theorem 1.2: abdi2025strongly
• Conjecture 1.3: schrijverobervation
• Theorem 1.4
• Definition 1.5: digraft
• Theorem 1.6
• Definition 1.7: basic digraft
• Definition 1.8: brick, brace
• Theorem 1.9
• Definition 1.10: elementary digraft