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

TL;DR

This paper introduces new techniques to improve lower bounds for the maximum induced acyclic subgraph size in digraphs, combining neighborhood refinement and variance analysis.

Contribution

It adapts existing bounds with neighborhood and variance-based refinements, providing tighter lower bounds for the problem.

Findings

A neighborhood-based refinement of the AGJS bound was proved.

Variance analysis of a randomized feedback vertex set yields a tighter bound.

The new bounds improve upon the original AGJS bound.

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.