Improved NP-Hardness of Approximation for Orthogonality Dimension and Minrank

TL;DR

This paper establishes new NP-hardness results for approximating the orthogonality dimension and minrank of graphs, showing these problems are hard to approximate within any constant factor, thus advancing understanding of their computational complexity.

Contribution

It proves the NP-hardness of approximating the orthogonality dimension and minrank within any constant factor for large parameters, improving previous hardness results.

Findings

NP-hardness of approximating orthogonality dimension over reals for large k

NP-hardness of approximating minrank related to index coding

Hardness results hold over finite fields as well

Abstract

The orthogonality dimension of a graph $G$ over $\mathbb{R}$ 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 $\mathsf{NP}$-hard to decide whether the orthogonality dimension of a given graph over $\mathbb{R}$ is at most $k$ or at least $2^{(1-o(1)) \cdot k/2}$. 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 $\mathsf{NP}$-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.