Parameterized Complexity of $(A,\ell)$-Path Packing

TL;DR

This paper investigates the parameterized complexity of the $(A, ext{ell})$-Path Packing problem, revealing complexity boundaries based on parameters like path length, size of $A$, and graph width measures, with some cases efficiently solvable and others NP-hard.

Contribution

The paper provides a detailed complexity landscape of the $(A, ext{ell})$-Path Packing problem across various parameters, including new polynomial-time and fixed-parameter tractability results.

Findings

Polynomial-time solvable for $ ext{ell} \, extless= 3$

NP-complete for constant $ ext{ell} \, extgreater= 4$

W[1]-hard for pathwidth + |A|, FPT for treewidth + $ ext{ell}$

Abstract

Given a graph $G = (V,E)$, $A \subseteq V$, and integers $k$ and $\ell$, the \textsc{$(A,\ell)$-Path Packing} problem asks to find $k$ vertex-disjoint paths of length $\ell$ that have endpoints in $A$ and internal points in $V \setminus A$. We study the parameterized complexity of this problem with parameters $|A|$, $\ell$, $k$, treewidth, pathwidth, and their combinations. We present sharp complexity contrasts with respect to these parameters. Among other results, we show that the problem is polynomial-time solvable when $\ell \le 3$, while it is NP-complete for constant $\ell \ge 4$. We also show that the problem is W[1]-hard parameterized by pathwidth${}+|A|$, while it is fixed-parameter tractable parameterized by treewidth${}+\ell$.