On the Confluence of Directed Graph Reductions Preserving Feedback Vertex Set Minimality

TL;DR

This paper explores graph reduction techniques that preserve minimal feedback vertex sets in directed graphs, focusing on properties like confluence to enable parallel processing and improve algorithm efficiency.

Contribution

It investigates the properties of reductions preserving minimal feedback vertex sets, especially the Church-Rosser property, and identifies the largest subset of such reductions for potential algorithmic benefits.

Findings

Identified reductions with the Church-Rosser property in directed graphs.

Analyzed the largest subset of reductions preserving minimal feedback vertex sets.

Discussed implications for parallelization and algorithm speed-up.

Abstract

In graph theory, the minimum directed feedback vertex set (FVS) problem consists in identifying the smallest subsets of vertices in a directed graph whose deletion renders the directed graph acyclic. Although being known as NP-hard since 1972, this problem can be solved in a reasonable time on small instances, or on instances having special combinatorial structure. In this paper we investigate graph reductions preserving all or some minimum FVS and focus on their properties, especially the Church-Rosser property, also called confluence. The Church-Rosser property implies the irrelevance of reduction order, leading to a unique directed graph. The study seeks the largest subset of reductions with the Church-Rosser property and explores the adaptability of reductions to meet this criterion. Addressing these questions is crucial, as it may impact algorithmic implications, allowing for parallelization and speeding up sequential algorithms.