Feedback Vertex Set on Geometric Intersection Graphs

TL;DR

This paper introduces an optimized algorithm for finding feedback vertex sets in unit disk graphs that improves previous methods and is optimal under the exponential-time hypothesis, with extensions to other geometric intersection graphs.

Contribution

It presents a faster, optimal algorithm for feedback vertex set on unit disk graphs, extending to fat objects without increasing complexity.

Findings

Runs in 2^{O(√k)}(n+m) time, improving previous algorithms.

Proves optimality assuming the exponential-time hypothesis.

Extends to geometric intersection graphs of fat objects.

Abstract

In this paper, we present an algorithm for computing a feedback vertex set of a unit disk graph of size $k$, if it exists, which runs in time $2^{O(\sqrt{k})}(n+m)$, where $n$ and $m$ denote the numbers of vertices and edges, respectively. This improves the $2^{O(\sqrt{k}\log k)}n^{O(1)}$-time algorithm for this problem on unit disk graphs by Fomin et al. [ICALP 2017]. Moreover, our algorithm is optimal assuming the exponential-time hypothesis. Also, our algorithm can be extended to handle geometric intersection graphs of similarly sized fat objects without increasing the running time.