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

TL;DR

This work studies Path Cover and Path Partition on trees and graphs with bounded treewidth. It delivers a linear-time algorithm for Path Cover on trees and a dynamic-programming scheme that solves Path Cover on graphs of treewidth $t$ in XP time $n^{t^{O(t)}}$ (or with a solution-size bound $kappa$, $kappa^{t^{O(t)}}n$), plus a randomized $2^{O(t)}n$ (deterministic $2^{O(t\log t)}n$) Cut&Count algorithm for Path Partition. The results extend to induced and/or edge-disjoint variants, providing a broad toolkit for path-cover-type problems beyond Hamiltonian paths. Overall, the paper advances both theoretical understanding and algorithmic practicalities for path covers on restricted graph classes, with clear implications for applications where treewidth is small.

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 $κ^{t^{O(t)}}n$ if there is an upper-bound $κ$ 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.

Figures (2)

• Figure 1: Illustration of the different types of a vertex with respect to some node $v$ and partial path. The set of types is $\mathcal{N}=\{\emptyset, \{-\},\{-,-\},\{\uparrow\},\{\downarrow\},\{\uparrow,\uparrow\},\{\downarrow,\downarrow\},\{\downarrow,\uparrow\},\{\uparrow,-\},\{\downarrow,-\}\}$. "$-$" shows a vertex with a neighbour inside the bag $X_v$, "$\uparrow$" shows a vertex with a neighbour in $G-G_v$, and "$\downarrow$" shows a vertex with a neighbour in $G_v-X_v$.
• Figure 2: A graph of arbitrarily large order $n$ and treewidth 2, with $\Theta(\sqrt{n})$ cut-vertices, and 2-connected components each of order $\Theta(\sqrt{n})$. An optimal solution to Path Cover consists of $\Theta(\sqrt{n})$ solution paths, each of length $\Theta(\sqrt{n})$, all going from the left to the right side of the graph. Thus, there is an unbounded number of solution paths going through every bag of any tree decomposition of bounded width.

Theorems & Definitions (22)

• Theorem 1
• definition 1
• definition 2
• proof : Proof of Theorem \ref{['thm:PCTL']}
• Theorem 2
• Corollary 3
• definition 3
• definition 4: kloks1994treewidth
• Lemma 4
• definition 5