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Asymptotics of the Minimal Feedback Arc Set in Erdős-Rényi Graphs

Harvey Diamond, Mark Kon, Louise Raphael

TL;DR

This paper studies the Minimal Feedback Arc Set problem on directed Erdős-Rényi graphs $D(n,p)$ by encoding feedforward and feedback arcs as $X$ and $Y$ counts above and below the diagonal of the adjacency matrix and comparing pre- and post-optimization ratios $X^*/Y^*$. It derives exponential tail bounds via Hoeffding, Bennett, and Chernoff-type inequalities to bound the probability that the optimized ratio exceeds $1+\epsilon$, showing this probability vanishes as $n\to\infty$ for fixed $p>0$ and also under sparse scaling $p\le C\log n/n$ with appropriate constants. The main finding is that the asymptotic ratio tends to 1 with high probability, implying little advantage from solving the FAS in large random digraphs across both dense and sparse regimes. The results provide explicit uniform large-deviation bounds and clarify the limits of FAS optimization in random structures, informing both theory and potential applications in network ordering tasks.

Abstract

Given a directed graph, the Minimal Feedback Arc Set (FAS) problem asks for a minimal set of arcs which, when removed, results in an acyclic graph. Equivalently, the FAS problem asks to find an ordering of the vertices that minimizes the number of feedback arcs. The FAS problem is considered an algorithmic problem of central importance in discrete mathematics. Our purpose in this paper is to consider the problem in the context of Erdős-Rényi random directed graphs, denoted $D(n,p)$, in which each possible directed arc is included with a fixed probability $p>0$. Our interest is the typical ratio of the number of feedforward arcs to the number of feedback arcs that are removed in the FAS problem. We show that as the number $n$ of vertices goes to infinity the probability that this ratio is greater than $1+ε$ for any fixed $ε> 0$ approaches zero. Similarly, letting $p$ go to zero as $n\rightarrow \infty$ this result remains true if $p>C\log{n}/n$ where $C$ depends on $ε$.

Asymptotics of the Minimal Feedback Arc Set in Erdős-Rényi Graphs

TL;DR

This paper studies the Minimal Feedback Arc Set problem on directed Erdős-Rényi graphs by encoding feedforward and feedback arcs as and counts above and below the diagonal of the adjacency matrix and comparing pre- and post-optimization ratios . It derives exponential tail bounds via Hoeffding, Bennett, and Chernoff-type inequalities to bound the probability that the optimized ratio exceeds , showing this probability vanishes as for fixed and also under sparse scaling with appropriate constants. The main finding is that the asymptotic ratio tends to 1 with high probability, implying little advantage from solving the FAS in large random digraphs across both dense and sparse regimes. The results provide explicit uniform large-deviation bounds and clarify the limits of FAS optimization in random structures, informing both theory and potential applications in network ordering tasks.

Abstract

Given a directed graph, the Minimal Feedback Arc Set (FAS) problem asks for a minimal set of arcs which, when removed, results in an acyclic graph. Equivalently, the FAS problem asks to find an ordering of the vertices that minimizes the number of feedback arcs. The FAS problem is considered an algorithmic problem of central importance in discrete mathematics. Our purpose in this paper is to consider the problem in the context of Erdős-Rényi random directed graphs, denoted , in which each possible directed arc is included with a fixed probability . Our interest is the typical ratio of the number of feedforward arcs to the number of feedback arcs that are removed in the FAS problem. We show that as the number of vertices goes to infinity the probability that this ratio is greater than for any fixed approaches zero. Similarly, letting go to zero as this result remains true if where depends on .
Paper Structure (6 sections, 7 theorems, 33 equations, 1 figure)

This paper contains 6 sections, 7 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Graphs with a fixed probability $p$ of an arc as $n \rightarrow \infty$
  3. Large deviation results with $p\rightarrow 0 \text{ as } n \rightarrow \infty$
  4. A Direct Inequality for $Pr\left(\textbf{X}-r\textbf{Y}\ge 0 \right)$
  5. Summary
  6. References

Key Result

Lemma 2.1

In an Erdős-Rényi graph on n vertices with probability $p$ of a given directed arc, the probability of obtaining a graph for which ${\frac{\textbf{X}^*}{\textbf{Y}^*} \ge r}$ in an FAS solution is no larger than $n!$ times the probability of obtaining a graph with an adjacency matrix satisfying ${\f

Figures (1)

  • Figure 1: Surface of the scaled difference on domain

Theorems & Definitions (11)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • ...and 1 more