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 ∪ V 2 , E ) , that is, pairs ( A , B ) where A ⊆ V 1 , B ⊆ V 2 , and A ∪ B are independent. These parameters also allow us to study bicliques in general graphs. When maximizing the cardinality | A ∪ B | , one finds the stability number α ( 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 ∪ 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 α 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 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 and Haemers in 2001 and Vallentin in 2020.