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Improved lower bounds for the maximum order of an induced acyclic subgraph

Shamil Asgarli, Donald Falkenhagen, Kaya Hoshi

TL;DR

This work extends Caro–Wei style lower bounds for maximum induced acyclic vertex sets from undirected graphs to digraphs by developing two complementary refinements of the AGJS bound: a neighborhood-based refinement and a variance-based bound. The neighborhood refinement applies the AGJS construction to a residual subgraph, producing a tight bound that leverages local degree structure, while the variance-based bound analyzes a randomized feedback vertex set (FVS) via the DL algorithm and uses the Bhatia–Davis inequality to obtain a second-order improvement. The paper provides exact expressions for the probability of a vertex not belonging to the DL output and derives a complete covariance catalog via inclusion–exclusion, enabling a closed-form variance term and a practical bound: $\alpha(D) \ge \sum_v \rho_D(v) + \dfrac{\mathrm{Var}(|S|)}{\sum_v \rho_D(v) - c}$. Numerical experiments on several random graph models show that the improvements are most pronounced in graphs with heterogeneous degree structure, while Erdős–Rényi digraphs exhibit little to no gain, illustrating the conditions under which each refinement is most effective.

Abstract

Computing the cardinality of a maximum induced acyclic vertex set in a digraph is NP-hard. Since finding an exact solution is computationally difficult, a fruitful approach is to establish high-quality lower bounds that are efficiently computable. We build on the Akbari--Ghodrati--Jabalameli--Saghafian (AGJS) bound for digraphs by adapting refinement techniques used by (a) Selkow and Harant--Mohr and (b) Angel--Campigotto--Laforest in their respective improvements of the Caro--Wei bound for undirected graphs. First, inspired by (a), we prove a neighborhood-based refinement of the AGJS bound that incorporates local degree data of each vertex. Second, inspired by (b), we compute the variance of the size of a feedback vertex set returned by a randomized algorithm. This result, combined with the Bhatia--Davis inequality, yields a tighter lower bound than the AGJS bound.

Improved lower bounds for the maximum order of an induced acyclic subgraph

TL;DR

This work extends Caro–Wei style lower bounds for maximum induced acyclic vertex sets from undirected graphs to digraphs by developing two complementary refinements of the AGJS bound: a neighborhood-based refinement and a variance-based bound. The neighborhood refinement applies the AGJS construction to a residual subgraph, producing a tight bound that leverages local degree structure, while the variance-based bound analyzes a randomized feedback vertex set (FVS) via the DL algorithm and uses the Bhatia–Davis inequality to obtain a second-order improvement. The paper provides exact expressions for the probability of a vertex not belonging to the DL output and derives a complete covariance catalog via inclusion–exclusion, enabling a closed-form variance term and a practical bound: . Numerical experiments on several random graph models show that the improvements are most pronounced in graphs with heterogeneous degree structure, while Erdős–Rényi digraphs exhibit little to no gain, illustrating the conditions under which each refinement is most effective.

Abstract

Computing the cardinality of a maximum induced acyclic vertex set in a digraph is NP-hard. Since finding an exact solution is computationally difficult, a fruitful approach is to establish high-quality lower bounds that are efficiently computable. We build on the Akbari--Ghodrati--Jabalameli--Saghafian (AGJS) bound for digraphs by adapting refinement techniques used by (a) Selkow and Harant--Mohr and (b) Angel--Campigotto--Laforest in their respective improvements of the Caro--Wei bound for undirected graphs. First, inspired by (a), we prove a neighborhood-based refinement of the AGJS bound that incorporates local degree data of each vertex. Second, inspired by (b), we compute the variance of the size of a feedback vertex set returned by a randomized algorithm. This result, combined with the Bhatia--Davis inequality, yields a tighter lower bound than the AGJS bound.

Paper Structure

This paper contains 18 sections, 13 theorems, 65 equations, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A neighborhood-based refinement for digraphs
  3. A variance-based approach via feedback vertex sets
  4. The DL algorithm and its correctness
  5. Structural properties of the DL algorithm
  6. Recovering the AGJS bound
  7. The variance of the DL algorithm
  8. A strategy via inclusion-exclusion
  9. A general ordering lemma
  10. Case 1: Nonadjacent vertices
  11. Case 2: Adjacent vertices with one arc
  12. Case 3: Adjacent vertices with two arcs
  13. The variance-based lower bound
  14. Neighborhood-based bound
  15. Variance-based bound
  16. ...and 3 more sections

Key Result

Lemma 2.1

Consider the function on the domain $x\geq 0$, $y\geq 0$, $0\leq z\leq \min(x,y)$. Suppose $a, b, c, d, s, t$ are nonnegative integers such that $a\leq c$, $b\leq d$, and $s\leq t$. Then $f(a,b,s) \geq f(c,d,t)$.

Theorems & Definitions (35)

  • Lemma 2.1: Monotonicity
  • proof
  • Theorem 2.2: Neighborhood-based refinement
  • proof
  • Definition 3.1: Right neighbors for a labeling
  • Definition 3.2: DL algorithm
  • Lemma 3.3: DL produces an FVS
  • proof
  • Lemma 3.4: Criticality of minimal FVS
  • proof
  • ...and 25 more