Disjoint paths problems are among the most prominent problems in combinatorial optimization. The edge- as well as vertex-disjoint paths problem, are NP-complete on directed and undirected graphs. But on undirected graphs, Robertson and Seymour (Graph Minors XIII) developed an algorithm for the vertex- and the edge-disjoint paths problem that runs in cubic time for every fixed number $p$ of terminal pairs, i.e. they proved that the problem is fixed-parameter tractable on undirected graphs. On directed graphs, Fortune, Hopcroft, and Wyllie proved that both problems are NP-complete already for $p=2$ terminal pairs. In this paper, we study the edge-disjoint paths problem (EDPP) on Eulerian digraphs, a problem that has received significant attention in the literature. Marx (Marx 2004) proved that the Eulerian EDPP is NP-complete even on structurally very simple Eulerian digraphs. On the positive side, polynomial time algorithms are known only for very restricted cases, such as $p\leq 3$ or where the demand graph is a union of two stars (see e.g. Ibaraki, Poljak 1991; Frank 1988; Frank, Ibaraki, Nagamochi 1995). The question of which values of $p$ the edge-disjoint paths problem can be solved in polynomial time on Eulerian digraphs has already been raised by Frank, Ibaraki, and Nagamochi (1995) almost 30 years ago. But despite considerable effort, the complexity of the problem is still wide open and is considered to be the main open problem in this area (see Chapter 4 of Bang-Jensen, Gutin 2018 for a recent survey). In this paper, we solve this long-open problem by showing that the Edge-Disjoint Paths Problem is fixed-parameter tractable on Eulerian digraphs in general (parameterized by the number of terminal pairs). The algorithm itself is reasonably simple but the proof of its correctness requires a deep structural analysis of Eulerian digraphs.

翻译：不交路径问题是组合优化中最突出的问题之一。边不交路径问题和点不交路径问题在有向图和无向图中均为NP完全。但在无向图中，Robertson和Seymour（图子式XIII）针对点不交和边不交路径问题提出了一种算法，对于任意固定数量的终端对$p$，该算法可在三次时间内运行，即证明了该问题在无向图中是固定参数可解的。在有向图中，Fortune、Hopcroft和Wyllie证明，当终端对数量$p=2$时，这两个问题已属NP完全。本文研究欧拉有向图中的边不交路径问题（EDPP），该问题在文献中备受关注。Marx（2004年）证明，即使在结构非常简单的欧拉有向图中，欧拉EDPP也是NP完全的。从积极方面看，目前已知的多项式时间算法仅适用于非常受限的情况，例如$p\leq 3$或需求图是两个星形图的并集（参见Ibaraki、Poljak 1991；Frank 1988；Frank、Ibaraki、Nagamochi 1995）。对于欧拉有向图中边不交路径问题能在多项式时间内求解的$p$值范围，Frank、Ibaraki和Nagamochi（1995年）早在近30年前就已提出该问题。但尽管付出了大量努力，该问题的复杂性至今仍悬而未决，并被视为此领域的主要开放问题（参见Bang-Jensen、Gutin 2018年综述第4章）。本文通过证明边不交路径问题在一般欧拉有向图中是固定参数可解的（以终端对数量为参数），解决了这一长期未决问题。该算法本身相对简洁，但其正确性证明需要对欧拉有向图进行深入的结构分析。