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 = (V1 U V2 , E), that is, pairs (A , B) where A c V1 , B c V2 , and A U B are independent. These parameters also allow us to study bicliques in general graphs. When maximizing the cardinality |A U B|, one finds the stability number alpha(G), wellknown to be polynomial -time computable. When maximizing the product |A | center dot | B |, one finds the parameter g(G), shown to be NP -hard by Peeters in 2003, and when maximizing the ratio |A | center dot |B|/ |A U 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 bal(G) (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lova ' sz & thetasym;-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.