Deterministic Single Exponential Time Algorithms for Co-Path Packing and Co-Path Set Parameterized by Treewidth

2026-05-19

AI Summary

This work addresses the induced path cover problem in graph editing—specifically, determining whether at most $k$ vertices (Co-Path Packing) or $k$ edges (Co-Path Set) can be removed so that the remaining graph becomes a disjoint union of induced paths. The paper presents the first deterministic single-exponential time algorithm parameterized by treewidth, achieving a running time of $O^*(c^{\text{tw}})$ for some constant $c$. By integrating dynamic programming, refined state compression, and subset convolution techniques over a tree decomposition, the approach efficiently avoids reliance on the randomized “Cut & Count” framework. This result matches the complexity bound of the best-known randomized algorithms and resolves a long-standing open question regarding derandomization for this problem.

Abstract

The \textsc{Co-Path Packing} (resp., \textsc{Co-Path Set}) problem asks whether a given graph can be edited to a collection of induced paths by deleting at most $k$ vertices (resp., $k$ edges). Both are fundamental problems with significant applications in bioinformatics and have been extensively studied within the framework of exact and parameterized algorithms. Currently, the state-of-the-art approach utilizes the randomized ``Cut \& Count'' technique, which solves \textsc{Co-Path Set} in $O^*(4^{\mathbf{tw}})$ time and \textsc{Co-Path Packing} in $O^*(5^{\mathbf{pw}})$ time, where $\mathbf{tw}$ is treewidth and $\mathbf{pw}$ is pathwidth. However, as there is no known method to derandomize the ``Cut \& Count'' technique, the existence of deterministic single exponential time algorithms for these problems parameterized by treewidth has remained an open question. In this paper, we resolve this gap by providing deterministic single exponential time algorithms for both problems when parameterized by treewidth.

Problem

• Co-Path Packing
• Co-Path Set
• treewidth
• deterministic algorithm
• single exponential time

Innovation

• deterministic algorithm
• single exponential time
• treewidth
• Co-Path Packing
• Co-Path Set