An Improved Deterministic Parameterized Algorithm for Cactus Vertex Deletion

TL;DR

This paper introduces a faster deterministic parameterized algorithm for Cactus Vertex Deletion, reducing the running time from 26^k to 17.64^k, and also improves algorithms for related problems like Even Cycle Transversal.

Contribution

It presents a novel deterministic algorithm with improved exponential running time for Cactus Vertex Deletion using measure and conquer analysis.

Findings

Reduced the algorithm's running time from 26^k to 17.64^k.

Applied the new algorithm to achieve a 17.64^k-time solution for Even Cycle Transversal.

Demonstrated the effectiveness of measure and conquer analysis with an elaborate measure.

Abstract

A cactus is a connected graph that does not contain $K_4 - e$ as a minor. Given a graph $G = (V, E)$ and integer $k \ge 0$, Cactus Vertex Deletion (also known as Diamond Hitting Set) is the problem of deciding whether $G$ has a vertex set of size at most $k$ whose removal leaves a forest of cacti. The current best deterministic parameterized algorithm for this problem was due to Bonnet et al. [WG 2016], which runs in time $26^kn^{O(1)}$, where $n$ is the number of vertices of $G$. In this paper, we design a deterministic algorithm for Cactus Vertex Deletion, which runs in time $17.64^kn^{O(1)}$. As a straightforward application of our algorithm, we give a $17.64^kn^{O(1)}$-time algorithm for Even Cycle Transversal. The idea behind this improvement is to apply the measure and conquer analysis with a slightly elaborate measure of instances.