Orienting the edges of an undirected graph such that the resulting digraph satisfies some given constraints is a classical problem in graph theory, with multiple algorithmic applications. In particular, an $st$-orientation orients each edge of the input graph such that the resulting digraph is acyclic, and it contains a single source $s$ and a single sink $t$. Computing an $st$-orientation of a graph can be done efficiently, and it finds notable applications in graph algorithms and in particular in graph drawing. On the other hand, finding an $st$-orientation with at most $k$ transitive edges is more challenging and it was recently proven to be NP-hard already when $k=0$. We strengthen this result by showing that the problem remains NP-hard even for graphs of bounded diameter, and for graphs of bounded vertex degree. These computational lower bounds naturally raise the question about which structural parameters can lead to tractable parameterizations of the problem. Our main result is a fixed-parameter tractable algorithm parameterized by treewidth.

翻译：将无向图的每条边定向，使得生成的有向图满足给定约束，是图论中的经典问题，具有多种算法应用。特别地，$st$-定向将输入图的每条边定向，使得生成的有向图是无环的，并且包含唯一的源点$s$和唯一的汇点$t$。计算图的$st$-定向可以高效完成，并在图算法特别是图绘图中具有重要应用。另一方面，寻找最多包含$k$条传递边的$st$-定向更具挑战性，且最近已被证明即使在$k=0$时也是NP难的。我们通过证明该问题即使对于有界直径的图和有界顶点度的图仍然是NP难的，强化了这一结果。这些计算下界自然地引出了一个问题：哪些结构参数能够导致该问题的可处理参数化。我们的主要结果是针对树宽参数化的固定参数可处理算法。