Optimization Problems over Unit-Distance Representations of Graphs

Details Optimization Problems over Unit-Distance Representations of Graphs Optimization Problems over Unit-Distance Representations of Graphs

2010-10-28

arxiv.org/abs/1010.6036v3

Mathematics

Optimization and Control

Scientific paper

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.

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