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

TL;DR

The study seeks the largest subset of reductions with the Church-Rosser property and explores the adaptability of reductions to meet this criterion, as it may impact algorithmic implications, allowing for parallelization and speeding up sequential algorithms.

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.

Figures (3)

• Figure 1: Illustration of the indiclique reduction. (a) $N_{G}^- (u)$ is a diclique, so $\textsc{indiclique}(u)$ is applicable. (b) We remove $u$ and all its incident arcs, and we add the new arcs $(p_i,s_j)$ for $i \in \{ 1,2,3 \}$ and $j \in \{ 1,2 \}$.
• Figure 2: Illustration of Lin and Jou reductions preconditions. (a) $\textsc{pie}$ is applicable on the blue and red arcs. Indeed, there no circuit going through the blue arcs $(b,v)$ and $(a,w)$ in $G$, and therefore in $G^\rightarrow$ and the same is true for the red arcs $(v,w)$ and $(w,u)$ in $G^\rightarrow$. Therefore we can remove the blue and red arcs from $G$. (b) $\textsc{core}(u)$ is applicable since $u$ and its neighbors, $\{u,u_1,u_2,u_3\}$ form a diclique. So we can add $\{u_1,u_2,u_3\}$ to the MFVS and remove them from $G$. (c) In the top the first case of $\textsc{dome}$ and in the bottom the second case. The arc $(u,v)$ is dominated and we can remove it from $G$.
• Figure 3: A digraph showing that the reductions in the article by Lin and Jou does not have the Church-Rosser property. If $G=(V,E)$ denotes the displayed digraph, the equalities $N_{G^\rightarrow}^- (c) = \{ a,b \} \subseteq \{ a,b,c,d \} = N_{G}^- (e)$ and $N_{G^\rightarrow}^- (d) = \{ a,c \} \subseteq \{ a,b,c,d \} = N_{G}^- (e)$ hold. Hence we can reduce $G$ using $\textsc{dome}(c,e)$ and $\textsc{dome}(d.e)$. We can apply $\textsc{dome}(d,e)$ followed by $\textsc{dome}(c,e)$. If we first apply $\textsc{dome}(c,e)$, however, the precondition of $\textsc{dome}(d,e)$ is not verified. Indeed, we have $N_{G - (c,e)}^- (d) = \{ a,c \} \not \subseteq \{ a,b,d \} = N_{G - (c,e)}^- (e)$ and the graph $(V, E - (c,e))$ cannot be reduced further.

Theorems & Definitions (5)

• Theorem 1: Sethi sethi
• Proposition 1
• Lemma 1
• Theorem 2
• proof