Graph and Election Problems Parameterized by Feedback Set Numbers

TL;DR

This paper explores the parameterized complexity of three graph modification problems related to feedback set parameters, with applications in social choice and biology, analyzing their computational difficulty based on various measures of acyclicity.

Contribution

It provides a detailed complexity analysis of three problems with respect to feedback set parameters, highlighting their computational challenges and applications.

Findings

Complexity results vary depending on feedback set parameters.

Some problems are fixed-parameter tractable under certain parameters.

Other problems remain computationally hard even with bounded feedback sets.

Abstract

This work investigates the parameterized complexity of three related graph modification problems. Given a directed graph, a distinguished vertex, and a positive integer k, Minimum Indegree Deletion asks for a vertex subset of size at most k whose removal makes the distinguished vertex the only vertex with minimum indegree. Minimum Degree Deletion is analogously defined, but deals with undirected graphs. Bounded Degree Deletion is also defined on undirected graphs, but has a positive integer d instead of a distinguished vertex as part of the input. It asks for a vertex subset of size at most k whose removal results in a graph in which every vertex has degree at most d. The first two problems have applications in computational social choice whereas the third problem is used in computational biology. We investigate the parameterized complexity with respect to the parameters "treewidth", "size of a feedback vertex set" and "size of a feedback edge set" respectively "size of a feedback arc set". Each of these parameters measures the "degree of acyclicity" in different ways.