Polynomial-time algorithms for PATH COVER and PATH PARTITION on trees and graphs of bounded treewidth

TL;DR

This paper presents efficient algorithms for PATH COVER and PATH PARTITION problems on trees and graphs with bounded treewidth, advancing understanding of these problems' computational complexity.

Contribution

It introduces a linear-time algorithm for PATH COVER on trees and polynomial-time algorithms for PATH COVER and PATH PARTITION on graphs of bounded treewidth, including improvements using Cut&Count.

Findings

PATH COVER solvable in linear time on trees.

Polynomial algorithms for PATH COVER and PATH PARTITION on bounded treewidth graphs.

Enhanced algorithms using Cut&Count technique for PATH PARTITION.

Abstract

In the PATH COVER problem, one asks to cover the vertices of a graph using the smallest possible number of (not necessarily disjoint) paths. While the variant where the paths need to be pairwise vertex-disjoint, which we call PATH PARTITION, is extensively studied, surprisingly little is known about PATH COVER. We start filling this gap by designing a linear-time algorithm for PATH COVER on trees. We show that PATH COVER can be solved in polynomial time on graphs of bounded treewidth using a dynamic programming scheme. It runs in XP time $n^{t^{O(t)}}$ (where $n$ is the number of vertices and $t$ the treewidth of the input graph) or $\kappa^{t^{O(t)}}n$ if there is an upper-bound $\kappa$ on the solution size. A similar algorithm gives an FPT $2^{O(t\log t)}n$ algorithm for PATH PARTITION, which can be improved to (randomized) $2^{O(t)}n$ using the Cut\&Count technique. These results also apply to the variants where the paths are required to be induced (i.e. chordless) and/or edge-disjoint.