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

TL;DR

This paper presents the first deterministic single exponential time algorithms for Co-Path Packing and Co-Path Set problems parameterized by treewidth, addressing a key open question in graph algorithms.

Contribution

The authors develop deterministic algorithms with single exponential time complexity for these problems, replacing previous randomized approaches.

Findings

Achieved deterministic algorithms for Co-Path Packing and Co-Path Set

Resolved the open problem of derandomizing algorithms parameterized by treewidth

Provided algorithms with single exponential time complexity

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.