Semidefinite approximations for bicliques and biindependent pairs

TL;DR

This work studies biindependent pairs in bipartite graphs and their connections to bicliques, introducing parameters $\alpha(G)$, $g(G)$, and $h(G)$ along with balanced variants. It establishes NP-hardness and NP-completeness results that motivate semidefinite programming relaxations and spectral bounds akin to the Lovász $\vartheta$-number, deriving first-order bounds $g_1(G)$ and $h_1(G)$ and their relationships to $\alpha(G)$. The authors develop closed-form eigenvalue bounds $\widehat{g}(G)$ and $\widehat{h}(G)$ for $\,r$-regular bipartite graphs, with equalities under strong symmetry, and connect these bounds to Hoffman's and Haemers' spectral bounds. They illustrate the bounds on classic graphs (perfect matchings, crown graphs, cycles, hypercubes) and discuss balanced variants via the Lasserre hierarchy, finding that symmetry often collapses the balanced relaxations to the unbalanced eigenvalue bounds. Overall, the paper provides a cohesive SDP and spectral framework to approximate and relate several graph parameters governing biindependent pairs and bicliques, with concrete complexity and structural insights and several open questions for balanced bounds.

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 $α(G)$, 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 $α_{\text{bal}}(G)$ (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lovász $\vartheta$-number, a well-known semidefinite bound on $α(G)$. 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).

Figures (3)

• Figure 1: Graph $F$, $\omega(F)=3$, 6 nodes, 10 edges.
• Figure 2: Graph $G$ with $\alpha(G)=3$, $\alpha_{\text{\rm bal}}(G)=2$, $h(G)=2/3$, and $g(G)=2$
• Figure 3: Bounds on $g_{\text{\rm bc}}(G)$; recall $h_1(B_0(\overline G))\le \widehat{h}(B_0(\overline G))$, with equality if $B_0(\overline G)$ is edge-transitive (Proposition \ref{['prop:hbbioneedgetransitive']}).

Theorems & Definitions (46)

• Lemma 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 2.1
• Corollary 2.2
• Definition 2.3
• Theorem 2.4
• Definition 2.5
• Lemma 2.6
• Corollary 2.7