Semidefinite approximations for bicliques and biindependent pairs

TL;DR

This paper introduces semidefinite programming bounds for graph parameters related to biindependent pairs in bipartite graphs, establishing their computational hardness and connecting them to spectral bounds like the Lovász theta number.

Contribution

It develops new semidefinite bounds for parameters g(G), h(G), and balanced independence number, extending spectral bounds and analyzing their computational complexity.

Findings

h(G) is NP-hard to compute

Deciding balanced maximum independent set is NP-complete

Semidefinite bounds generalize Lovász theta number

Abstract

We investigate some graph parameters dealing with biindependent pairs $(A,B)$ in a bipartite graph $G=(V_1\cup V_2,E)$, i.e., pairs $(A,B)$ where $A\subseteq V_1$, $B\subseteq V_2$ and $A\cup B$ is independent. These parameters also allow to study bicliques in general graphs. When maximizing the cardinality $|A\cup B|$ one finds the stability number $\alpha(G)$, well-known to be polynomial-time computable. When maximizing the product $|A|\cdot |B|$ one finds the parameter $g(G)$, shown to be NP-hard by Peeters (2003), and when maximizing the ratio $|A|\cdot |B|/|A\cup B|$ one finds $h(G)$, introduced by Vallentin (2020) for bounding product-free sets in finite groups. We show that $h(G)$ is an NP-hard parameter and, as a crucial ingredient, that it is NP-complete to decide whether a bipartite graph $G$ has a balanced maximum independent set. These hardness results motivate introducing semidefinite programming bounds for $g(G)$, $h(G)$, and $\alpha_{\text{bal}}(G)$ (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lov\'{a}sz $\vartheta$-number, a well-known semidefinite bound on $\alpha(G)$. In addition we formulate closed-form eigenvalue bounds and we show relationships among them as well as with earlier spectral parameters by Hoffman, Haemers (2001) and Vallentin (2020).