Semidefinite Approximations for Bicliques and Bi-Independent Pairs

We investigate some graph parameters dealing with bi-independent pairs (A, B) in a bipartite graph $G=(V_1\cup V_2,E)$, that is, pairs (A, B) where $A\subseteq V_1,\hspace{0.17em}B\subseteq V_2$, and $A\cup B$ are independent. These parameters also allow us 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 $|A|·|B|$, one finds the parameter g(G), shown to be NP-hard by Peeters in 2003, and when maximizing the ratio $|A|·|B|/|A\cup B|$, one finds h(G), introduced by Vallentin in 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 (SDP) 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 Lovász ϑ-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 and Haemers in 2001 and Vallentin in 2020.

Funding: This work was supported by H2020 Marie Skłodowska-Curie Actions [Grant 813211 (POEMA)].