Optimization Problems over Unit-Distance Representations of Graphs

TL;DR

This paper explores the mathematical relationship between unit-distance graph representations, Lovasz theta number, and hypersphere numbers, introducing new min-max theorems and generalizations with computational complexity insights.

Contribution

It establishes new min-max theorems connecting graph representations and Lovasz theta, and generalizes hypersphere numbers to ellipsoids, revealing NP-hardness of related problems.

Findings

Derived min-max theorems linking graph representations and Lovasz theta.

Introduced a generalized hypersphere number involving ellipsoids.

Proved NP-hardness of optimization problems over positive semidefinite forms.

Abstract

We study the relationship between unit-distance representations and Lovasz theta number of graphs, originally established by Lovasz. We derive and prove min-max theorems. This framework allows us to derive a weighted version of the hypersphere number of a graph and a related min-max theorem. Then, we connect to sandwich theorems via graph homomorphisms. We present and study a generalization of the hypersphere number of a graph and the related optimization problems. The generalized problem involves finding the smallest ellipsoid of a given shape which contains a unit-distance representation of the graph. We prove that arbitrary positive semidefinite forms describing the ellipsoids yield NP-hard problems.