Approximating the Orthogonality Dimension of Graphs and Hypergraphs
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A t-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in R^t to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph H, denoted by ξ(H), is the smallest integer t for which there exists a t-dimensional orthogonal representation of H. In this paper we study computational aspects of the orthogonality dimension of graphs and hypergraphs. We prove that for every k ≥ 4, it is NP-hard (resp. quasi-NP-hard) to distinguish n-vertex k-uniform hypergraphs H with ξ(H) ≤ 2 from those satisfying ξ(H) ≥Ω(log^δ n) for some constant δ>0 (resp. ξ(H) ≥Ω(log^1-o(1) n)). For graphs, we relate the NP-hardness of approximating the orthogonality dimension to a variant of a long-standing conjecture of Stahl. We also consider the algorithmic problem in which given a graph G with ξ(G) ≤ 3 the goal is to find an orthogonal representation of G of as low dimension as possible, and provide a polynomial time approximation algorithm based on semidefinite programming.
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