Asymptotic Lower Bounds for the Feedback Arc Set Problem in Random Graphs

Given a directed graph, the Minimum Feedback Arc Set (FAS) problem asks for a minimum (size) set of arcs in a directed graph, which, when removed, results in an acyclic graph. In a seminal paper, Berger and Shor [1], in 1990, developed initial upper bounds for the FAS problem in general directed graphs. Here we find asymptotic \textit{lower bounds} for the FAS problem in a class of random, oriented, directed graphs derived from the Erdős-Rényi model $G(n,M)$, with n vertices and M (undirected) edges, the latter randomly chosen. Each edge is then randomly given a direction to form our directed graph. We show that $$Pr\left(\textbf{Y}^* \le M \left( \frac{1}{2} -\sqrt{\frac{\log n}{Δ_{av}}}\right)\right)$$ approaches zero exponentially in $n$, with $\textbf{Y}^$ the (random) size of the minimum feedback arc set and $Δ_{av}=2M/n$ the average vertex degree. Lower bounds for random tournaments, a special case, were obtained by Spencer [12] and de la Vega [13] and these are discussed. In comparing the bound above to averaged experimental FAS data on related random graphs developed by K. Hanauer [7] we find that the approximation $\textbf{Y}^{av} \approx M\left( \frac{1}{2} -\frac{1}{2}\sqrt{\frac{\log n}{Δ{av}}}\right)$ lies remarkably close graphically to the algorithmically computed average size $\textbf{Y}^*_{av}$ of minimum feedback arc sets.

💡 Research Summary

The paper addresses the Minimum Feedback Arc Set (FAS) problem, which asks for the smallest set of arcs whose removal makes a directed graph acyclic. While upper bounds and algorithmic results for general digraphs have been extensively studied (notably by Berger and Shor in 1990), lower bounds—especially for random directed graphs—have received far less attention. The authors focus on a natural random model derived from the Erdős‑Rényi G(n,M) construction: first choose M undirected edges uniformly at random among n vertices, then orient each edge independently with probability ½ in either direction. This yields an “oriented” digraph without two‑cycles.

The key random variable is Y, the number of feedback arcs (entries below the diagonal) in a fixed vertex ordering. Because each of the M edges is oriented independently, Y follows a binomial distribution B(M,½). The minimum possible number of feedback arcs over all n! vertex permutations is denoted Y*. The authors observe that the event {Y* ≤ k} can only occur if there exists at least one permutation for which Y ≤ k. Consequently, Lemma 2.1 gives the simple bound
 Pr(Y* ≤ k) ≤ n!·Pr(Y ≤ k).

To control Pr(Y ≤ k) they apply Hoeffding’s inequality to the sum of independent Bernoulli variables, obtaining Theorem 2.2:
 Pr(Y ≤ M(½ – t)) ≤ exp(–2Mt²) for any t > 0.
Combining the two results yields
 Pr(Y* ≤ M(½ – t)) ≤ n!·exp(–2Mt²).

The authors then choose t as a function of the average degree Δ_av = 2M/n, namely
 t = r·log n / Δ_av,
where r is a positive constant. Using Stirling’s approximation for n! (log n! ≤ n log n – n + O(log n)), the exponential term n!·exp(–2Mt²) simplifies to a bound of order 3√n e^(–n), which decays exponentially in n. Hence, with probability tending to 1 exponentially fast,
 Y* ≥ M(½ – r·log n / Δ_av).

A direct corollary follows: if n log n / M → 0 (i.e., the graph is sufficiently dense), then for any ε > 0,
 Pr(Y* ≥ M(½ – ε)) → 1.
This reproduces, in a probabilistic setting, the classical Erdős–Moon result that almost all dense oriented graphs require roughly half of their arcs to be removed to become acyclic.

The paper also discusses the special case of random tournaments (M = n choose 2). Known lower bounds by Spencer and de la Vega state that the probability that a tournament’s minimum feedback arc set is smaller than M/2 – O(n^{3/2}) tends to zero. The authors’ general bound reduces to a comparable statement when Δ_av ≈ n, confirming consistency with prior work.

Empirical validation is performed using data from K. Hanauer’s master’s thesis, which contains average minimum feedback arc set sizes computed by several exact and heuristic algorithms on random oriented graphs of varying n and edge densities. The observed average size (\mathbf{Y}^{}_{av}) aligns closely with the heuristic formula
 (\mathbf{Y}^{}{av} ≈ M\bigl(½ – ½\sqrt{\log n / Δ{av}}\bigr)).
This formula is a slight relaxation of the theoretical lower bound, yet the graphical plots show an almost perfect match across a broad range of parameters, especially for denser graphs.

In summary, the contributions of the paper are threefold:

1. It provides a clean, probabilistic lower bound for the minimum feedback arc set size in the widely studied G(n,M) oriented model.
2. It demonstrates that a combination of Hoeffding’s concentration inequality and a union bound over all vertex permutations yields a powerful yet elementary technique for such problems.
3. It bridges theory and practice by comparing the bound with extensive experimental data, confirming that the bound is not merely asymptotic but also predictive for moderate graph sizes.

The authors acknowledge limitations: the bound becomes trivial when the graph is very sparse (M = O(n)), because the term r·log n / Δ_av may exceed ½. They suggest future work could explore tighter tail inequalities, alternative random models such as D(n,p), and refined combinatorial arguments to obtain meaningful lower bounds in the sparse regime. Additionally, investigating the gap between the derived lower bound and known upper bounds could lead to improved approximation algorithms for the FAS problem in random settings.