Preprocessing to Reduce the Search Space: Antler Structures for Feedback Vertex Set

TL;DR

This paper introduces antler structures for Feedback Vertex Set, a preprocessing technique that reduces search space and enhances fixed-parameter tractable algorithms for solving NP-hard problems efficiently.

Contribution

It proposes a novel antler decomposition for Feedback Vertex Set and develops fixed-parameter algorithms based on this structure, advancing preprocessing methods for NP-hard problems.

Findings

Antler decomposition identifies vertices in optimal solutions.

Preprocessing with antler structures reduces search space.

Algorithms using antler structures solve Feedback Vertex Set more efficiently.

Abstract

The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly. We explore this direction for the classic Feedback Vertex Set problem on undirected graphs, leading to a new type of graph structure called antler decomposition, which identifies vertices that belong to an optimal solution. It is an analogue of the celebrated crown decomposition which has been used for Vertex Cover. We develop the graph structure theory around such decompositions and develop fixed-parameter tractable algorithms to find them, parameterized by the number of vertices for which they witness presence in an optimal solution. This reduces the search space of fixed-parameter tractable algorithms parameterized by the solution size that solve Feedback Vertex Set.