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Autre Publication Scientifique Année : 2023

Semidefinite approximations for bicliques and biindependent pairs

Résumé

We investigate some graph parameters asking to maximize the size of biindependent pairs $(A,B)$ in a bipartite graph $G=(V_1\cup V_2,E)$, 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 (via bipartite double graphs). When the size is the number $|A\cup B|$ of vertices one finds the stability number $\alpha(G)$, well-known to be polynomial-time computable. When the size is the product $|A|\cdot |B|$ one finds the parameter $g(G)$, shown to be NP-hard by Peeters (2003), and when the size is the ratio $|A|\cdot |B|/|A\cup |B|$ one finds the parameter $h(G)$, 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\'asz $\vartheta$-number, a well-known semidefinite bound on $\alpha(G)$ (equal to it for $G$ bipartite). In addition we formulate closed-form eigenvalue bounds, which coincide with the semidefinite bounds for vertex- and edge-transitive graphs, and we show relationships among them as well as with earlier spectral parameters by Hoffman, Haemers (2001) and Vallentin (2020).

Dates et versions

hal-04074144 , version 1 (19-04-2023)

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Monique Laurent, Sven Polak, Luis Felipe Vargas. Semidefinite approximations for bicliques and biindependent pairs. 2023. ⟨hal-04074144⟩
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