Approximating the Orthogonality Dimension of Graphs and Hypergraphs

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 overline{xi}(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 overline{xi}(H) = Omega(log^delta n) for some constant delta>0 (resp. overline{xi}(H) >= Omega(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 overline{xi}(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

Improved NP-Hardness of Approximation for Orthogonality Dimension and Minrank

The orthogonality dimension of a graph G over ? is the smallest integer k for which one can assign a nonzero k-dimensional real vector to each vertex of G, such that every two adjacent vertices receive orthogonal vectors. We prove that for every sufficiently large integer k, it is NP-hard to decide whether the orthogonality dimension of a given graph over ? is at most k or at least 2^{(1-o(1))?k/2}. We further prove such hardness results for the orthogonality dimension over finite fields as well as for the closely related minrank parameter, which is motivated by the index coding problem in information theory. This in particular implies that it is NP-hard to approximate these graph quantities to within any constant factor. Previously, the hardness of approximation was known to hold either assuming certain variants of the Unique Games Conjecture or for approximation factors smaller than 3/2. The proofs involve the concept of line digraphs and bounds on their orthogonality dimension and on the minrank of their complement

The (Generalized) Orthogonality Dimension of (Generalized) Kneser Graphs: Bounds and Applications

The orthogonality dimension of a graph $G=(V,E)$ over a field $\mathbb{F}$ is the smallest integer $t$ for which there exists an assignment of a vector $u_v \in \mathbb{F}^t$ with $\langle u_v,u_v \rangle \neq 0$ to every vertex $v \in V$, such that $\langle u_v, u_{v'} \rangle = 0$ whenever $v$ and $v'$ are adjacent vertices in $G$. The study of the orthogonality dimension of graphs is motivated by various application in information theory and in theoretical computer science. The contribution of the present work is two-folded. First, we prove that there exists a constant $c$ such that for every sufficiently large integer $t$, it is $\mathsf{NP}$-hard to decide whether the orthogonality dimension of an input graph over $\mathbb{R}$ is at most $t$ or at least $3t/2-c$. At the heart of the proof lies a geometric result, which might be of independent interest, on a generalization of the orthogonality dimension parameter for the family of Kneser graphs, analogously to a long-standing conjecture of Stahl (J. Comb. Theo. Ser. B, 1976). Second, we study the smallest possible orthogonality dimension over finite fields of the complement of graphs that do not contain certain fixed subgraphs. In particular, we provide an explicit construction of triangle-free $n$-vertex graphs whose complement has orthogonality dimension over the binary field at most $n^{1-\delta}$ for some constant $\delta >0$. Our results involve constructions from the family of generalized Kneser graphs and they are motivated by the rigidity approach to circuit lower bounds. We use them to answer a couple of questions raised by Codenotti, Pudl\'{a}k, and Resta (Theor. Comput. Sci., 2000), and in particular, to disprove their Odd Alternating Cycle Conjecture over every finite field.Comment: 19 page