Edge-Disjoint Paths in Eulerian Digraphs

TL;DR

This paper settles the long-standing open problem of fixed-parameter tractability for the edge-disjoint paths problem on Eulerian digraphs, parameterized by the number p of terminal pairs. The authors develop a near-linear strategy that reduces arbitrary instances to degree-4 graphs away from terminals, then leverages large cylindrical walls to either extract large routers or flat swirls. Routers yield irrelevancy results via Frank-type reductions, while flat swirls are tamed and flattened to enable embedding and further reductions, ultimately bounding the problem by a function f(p) times a polynomial in the input size. The core innovations include the development of routers and swirls as routing devices, the introduction of coastal maps for structured islands, and a comprehensive incidence-graph framework to handle edge-disjoint routing with edge- rather than vertex-disjoint constraints. The combination of structural graph theory with algorithmic reductions yields an fpt-time algorithm for EDPP on Eulerian digraphs, marking a significant step forward in directed disjoint-path problems and their parameterized complexity, with potential implications for related routing and flow problems on Eulerian structures.

Abstract

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.

Figures (16)

• Figure 1: Cylindrical grid $G_4$ and the cylindrical wall of order $4$. The perimeters of the wall are depicted using thick edges.
• Figure 2: Wall with wall-coordinates. The paths $\mathcal{P}$ forming a matching are marked in red. The green area is a subwall, the blue area forms a band, and the intersection of both areas, marked in dark blue, is a tile.
• Figure 3: The left side is a figure of a $3$-swirl in the wall. The right figure represents a swirl grasped by a tile.
• Figure 4: An example of three attachments to a tile containing strongly planar vertices and possible non-planarities, neither of which induce a usable cross in the tile.
• Figure 5: The left side depicts of a wall with two jump-sequences marked in red and orange. The right hand side depicts the associated jump-paths to the two jump-sequences where one of which is a jump cycle.

Theorems & Definitions (256)

• Definition 1.1: Edge-Disjoint Paths Problem
• Theorem 1.2
• Definition 3.1: Eulerian Digraph
• Definition 3.2: Paths and cycles
• Definition 3.3: Linkages
• Remark 3.4
• Definition 3.5: $A{-}B$-linkage
• Theorem 3.6
• Definition 3.7: $G\cup H$, $G\cap H$ and $G\setminus H$
• Remark 3.8