Revised: August 21, 2022

Published: December 29, 2022

**Keywords:**orthogonality dimension, minrank, Kneser graphs, hardness of approximation, circuit complexity

**Categories:**complexity theory, hardness of approximation, circuit complexity, orthogonality dimension, minrank, Kneser graphs

**ACM Classification:**F.2.2, G.2.2

**AMS Classification:**05C50, 68Q06, 68Q17, 68R10

**Abstract:**
[Plain Text Version]

The *orthogonality dimension* of a graph $G=(V,E)$ over a field $\Fset$ is the smallest integer $t$ for which there exists an assignment of a vector $u_v \in \Fset^t$ with $\langle u_v,u_v \rangle \neq 0$ to every vertex $v \in V$, such that
$\langle u_v, u_{w} \rangle = 0$ whenever $v$ and $w$
are adjacent vertices in $G$.
The study of the orthogonality dimension of graphs is motivated by various applications in information theory and in theoretical computer science.
The contribution of the present work is
twofold.

First, we prove that there exists a constant $c$ such that for every sufficiently large
fixed
integer $t$, it is $\NP$-hard to
distinguish between the cases that the orthogonality dimension over $\R$ of an input graph is at most $t$ and 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
(where vertices correspond to *subspaces* rather than vectors)
for the family of *Kneser graphs*. The result is related to a long-standing graph-theoretic conjecture due to Stahl (J. Combin. Theory--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
a family of
*generalized Kneser graphs*
(namely, threshold Kneser graphs)
and are motivated by the rigidity approach to circuit lower bounds.
We use them to answer
questions raised by Codenotti, Pudlák, and Resta (Theoret. Comput. Sci., 2000), and in particular, to disprove their Odd Alternating Cycle Conjecture over every finite field.

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A conference version of this paper appeared in the