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

2025-11-10

AI Summary

This paper studies the PATH COVER problem—covering all vertices of a graph with the minimum number of (not necessarily vertex-disjoint) paths. Historically overlooked, especially for trees and bounded-treewidth graphs, it lacked efficient exact algorithms. We present the first linear-time exact algorithm for trees (O(n)) and a polynomial-time algorithm for graphs of treewidth t running in n^{t^{O(t)}} time. For the related PATH PARTITION problem, we improve the randomized running time to 2^{O(t)}n. Our approach unifies these results under a tree-decomposition-based dynamic programming framework, enhanced by the Cut&Count technique to accelerate state merging. The framework naturally extends to variants imposing constraints such as induced paths or edge-disjointness. These contributions fill a theoretical gap in non-vertex-disjoint path covering, significantly advancing both computational efficiency and applicability to structured graph classes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Topic discovery and tracking

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 $2^{O(tlog t)}n$ 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.

Problem

• Develop linear-time algorithm for PATH COVER on trees
• Design polynomial-time solution for PATH COVER on bounded treewidth graphs
• Provide FPT algorithms for PATH PARTITION using dynamic programming

Innovation

• Linear-time algorithm for PATH COVER on trees
• Polynomial-time dynamic programming for bounded treewidth
• FPT algorithms using Cut&Count technique