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. That is, $\mathcal{B}$ 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 the previous 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 orientations can be solved in deterministic polynomial time. Along the way, we discover appealing connections to the theory of perfect matching lattices.

翻译：设$\vec{G}=(V,E^+\cup E^-)$为一个双向图，其底层无向图$G=(V,E)$是2-边连通的。**强连通定向**定义为包含每条边$e\in E$恰好一个方向$e^+$或$e^-$的弧子集，且该子集在$\vec{G}$中诱导出一个强连通子图。给定一个由$V$的真子集构成的族$\mathcal{F}$，若对于每个$U\in\mathcal{F}$，恰好有一条弧进入$U$，则称该强连通定向是**紧的**。我们给出一个多项式时间算法，用于构造一个由紧定向组成的集合$\mathcal{B}$，该集合构成紧定向线性包络的整基。即$\mathcal{B}$是紧定向的一个线性无关子集，且紧定向线性包络中的每个整数向量均可表示为$\mathcal{B}$的整数线性组合。这推广了Abdi、Conuéjols、Liu和Silina（IPCO 2025）的主要结果，他们在一等价设定中非构造性地证明了此类基的存在性。先前证明使用了多面体理论，而我们的证明纯组合化且能导出多项式时间算法。作为算法的一个应用，我们证明了带奇偶约束的紧强连通定向问题可在确定多项式时间内求解。在此过程中，我们发现了与完美匹配格理论的巧妙联系。