Constant Approximating Disjoint Paths on Acyclic Digraphs is W[1]-hard

TL;DR

This paper proves that approximating the Max Disjoint Paths problem on acyclic digraphs within any constant factor is W[1]-hard, highlighting the problem's computational difficulty in parameterized complexity.

Contribution

It establishes the first non-trivial parameterized hardness of approximation for Max Disjoint Paths on acyclic graphs, using a novel self-reduction and probabilistic combinatorial construction.

Findings

Max Disjoint Paths is W[1]-hard to approximate within any constant factor on acyclic digraphs.

The proof introduces an elementary self-reduction guided by a probabilistically constructed combinatorial object.

This result advances understanding of the parameterized complexity of approximation problems in directed acyclic graphs.

Abstract

In the Disjoint Paths problem, one is given a graph with a set of $k$ vertex pairs $(s_i,t_i)$ and the task is to connect each $s_i$ to $t_i$ with a path, so that the $k$ paths are pairwise disjoint. In the optimization variant, Max Disjoint Paths, the goal is to maximize the number of vertex pairs to be connected. We study this problem on acyclic directed graphs, where Disjoint Paths is known to be W[1]-hard when parameterized by $k$. We show that in this setting Max Disjoint Paths is W[1]-hard to $c$-approximate for any constant $c$. To the best of our knowledge, this is the first non-trivial result regarding the parameterized approximation for Max Disjoint Paths with respect to the natural parameter $k$. Our proof is based on an elementary self-reduction that is guided by a certain combinatorial object constructed by the probabilistic method.