Parameterizing Path Partitions

TL;DR

This paper investigates the computational complexity of partitioning graphs into a small number of paths, exploring various variants and parameters, and establishing both hardness results and fixed-parameter tractability under different conditions.

Contribution

It provides new complexity results for path partition problems, including hardness on specific graph classes and fixed-parameter algorithms based on neighborhood diversity and other parameters.

Findings

NP-hardness on planar bipartite DAGs and undirected bipartite graphs

W[1]-hardness parameterized by number of paths on DAGs

FPT algorithms for neighborhood diversity and vertex cover parameters

Abstract

We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The Path Partition problem (PP) has been studied extensively, as it includes Hamiltonian Path as a special case. The natural variants where the paths are required to be either \emph{induced} (Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition, SPP), have received much less attention. Both problems are known to be NP-complete on undirected graphs; we strengthen this by showing that they remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP remains NP-hard on undirected bipartite graphs. When parameterized by the natural parameter ``number of paths'', both SPP and IPP are shown to be W[1]-hard on DAGs. We also show that SPP is in XP both for DAGs and undirected graphs for the same parameter, as well as for other special subclasses of directed graphs (IPP is known to be NP-hard on undirected graphs, even for two paths). On the positive side, we show that for undirected graphs, both problems are in FPT, parameterized by neighborhood diversity. We also give an explicit algorithm for the vertex cover parameterization of PP. When considering the dual parameterization (graph order minus number of paths), all three variants, IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the mentioned neighborhood diversity and dual parameterization results to directed graphs; here, we need to define a proper novel notion of directed neighborhood diversity. As we also show, most of our results transfer to the case of covering by edge-disjoint paths, and purely covering.