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On the complexity of the disjoint paths problem. (English) Zbl 0770.68072
Summary: We consider the disjoint paths problem. Given a graph \(G\) and a subset \(S\) of the edge-set of \(G\) the problem is to decide whether there exists a family \(\mathcal F\) of disjoint circuits in \(G\) each containing exactly one edge of \(S\) such that every edge in \(S\) belongs to a circuit in \(\mathcal C\). By a well-known theorem of P. Seymour [On odd cuts and plane multicommodity flows, Proc. London Math. Soc. (3)42, 178-192 (1981; Zbl 0447.90026)] the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphs \(G\). We show that (assuming \(P \neq NP\)) one can drop neither planarity nor the Eulerian condition on \(G\) without losing polynomial time solvability. We prove the NP-completeness of the planar edge-disjoint paths problem by showing the NP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assuming \(P \neq NP\)) a conjecture of A. Schrijver [Homotopic Routing Methods, in: Paths, Flows and VLSI Layout, Algorithms Comb. 9, 329-371 (1990; Zbl 0732.90087)] concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjugate of A. Frank mentioned in A. Sebö [Dual Integrality and Multicommodity Flows. Combinatorics, Colloquia Mathematica Societatis János Bolyai, 52, 453-469 (1988; Zbl 0695.90041)]. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minor-closed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show the NP-completeness of the half- integral relaxation of the edge-disjoint paths problem. This implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerian graphs.

MSC:
68Q25 Analysis of algorithms and problem complexity
68R10 Graph theory (including graph drawing) in computer science
05C38 Paths and cycles
05C45 Eulerian and Hamiltonian graphs
05C10 Planar graphs; geometric and topological aspects of graph theory
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