Edge-Disjoint Paths in Eulerian Digraphs
CoRR(2024)
摘要
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≤ 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.
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