Optimization Problems over Unit-Distance Representations of Graphs
Optimization and Control
2018-01-30 v4 Combinatorics
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.
Cite
@article{arxiv.1010.6036,
title = {Optimization Problems over Unit-Distance Representations of Graphs},
author = {Marcel K. de Carli Silva and Levent Tunçel},
journal= {arXiv preprint arXiv:1010.6036},
year = {2018}
}