On the Parameterized Tractability of Packing Vertex-Disjoint A-Paths with Length Constraints

TL;DR

This paper investigates the computational complexity of packing vertex-disjoint A-paths with length constraints, providing hardness results and fixed-parameter tractable algorithms based on various structural graph parameters.

Contribution

It establishes W[1]-hardness for the problem with certain parameters and offers FPT algorithms and kernelization results for others, advancing understanding of parameterized complexity in path packing.

Findings

W[1]-hardness when parameterized by distance to path and |A|

FPT algorithms for combined parameters distance to cluster + |A| and distance to cluster + ll

Kernel with O(vc^2) vertices for vertex cover number parameter

Abstract

Given an undirected graph G and a set A \subseteq V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G) \setminus A. An A-path is called an (A,\ell)-path if the length of the path is exactly \ell. In the {\sc (A, \ell)-Path Packing} problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, \ell)-paths in G or not. We pursue this problem with respect to structural parameters. We prove that ALPP is W[1]-hard when it is parameterized by the combined parameter distance to path (dtp) and |A|. In addition, we consider the combined parameters distance to cluster (cvd) + |A| and distance to cluster (cvd) + \ell. For both these combined parameters, we provide FPT algorithms. Finally, we consider the vertex cover number (vc) as the parameter and provide a kernel with O(vc^2) vertices.