Disjoint paths in tournaments

TL;DR

This paper proves that for any fixed number of vertex-disjoint paths, there exists a polynomial-time algorithm to determine their existence in semicomplete digraphs, extending known results from the case of two paths.

Contribution

The paper generalizes the polynomial-time solvability of the disjoint paths problem from two paths to any fixed number in semicomplete digraphs.

Findings

Polynomial-time algorithm for fixed k disjoint paths in semicomplete digraphs

Extension of previous results from k=2 to all fixed k

Addresses NP-completeness in general digraphs for k ≥ 2

Abstract

Given $k$ pairs of vertices $(s_i,t_i)$, $1\le i\le k$, of a digraph $G$, how can we test whether there exist $k$ vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but for $k=2$ there is a polynomial-time algorithm when $G$ is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen. Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $G$ is semicomplete.