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

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.