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# Approximating the Orthogonality Dimension of Graphs and Hypergraphs

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LIPIcs.MFCS.2019.39.pdf
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## Acknowledgements

We are grateful to Alexander Golovnev for useful discussions and to the anonymous reviewers for their valuable suggestions.

## Cite As

Ishay Haviv. Approximating the Orthogonality Dimension of Graphs and Hypergraphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.MFCS.2019.39

## Abstract

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) <= 2 from those satisfying 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph coloring
• Theory of computation → Problems, reductions and completeness
• Theory of computation → Approximation algorithms analysis
##### Keywords
• orthogonal representations of hypergraphs
• orthogonality dimension
• hardness of approximation
• Kneser and Schrijver graphs
• semidefinite programming

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