The Complexity of Packing Edge-Disjoint Paths

TL;DR

This paper investigates the computational complexity of Path Packing, a generalization of Hamiltonian Path, providing fixed-parameter algorithms, hardness results, and exploring special cases with varying path lengths and coverage constraints.

Contribution

It introduces new fixed-parameter algorithms for Path Packing, establishes tight complexity bounds, and analyzes special cases including path length restrictions and edge coverage.

Findings

FPT algorithm for Path Packing on trees with the number of paths as parameter

NP-hardness for packing paths of length three

Polynomial-time solvability for packing paths of length two

Abstract

We introduce and study the complexity of Path Packing. Given a graph $G$ and a list of paths, the task is to embed the paths edge-disjoint in $G$. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard. Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case EPC where the paths have to cover every edge in $G$ exactly once is already NP-hard for two paths on 4-regular graphs.