Abstract
We establish that the extension complexity of the \(n\times n\) correlation polytope is at least \(1.5\,^n\) by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least \(1.5^n\), and thus the nondeterministic communication complexity of the unique-disjointness predicate is at least \(.58n\). We thereby slightly improve on the previously best known lower bounds \(1.24^n\) and \(.31n\), respectively.
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Acknowledgments
We thank Yuri Faenza and Kanstantsin Pashkovich for their helpful remarks on an earlier version of this paper. Part of this research was funded by the German Research Foundation (DFG): “Extended Formulations in Combinatorial Optimization” (KA 1616/4-1).
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Kaibel, V., Weltge, S. A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially. Discrete Comput Geom 53, 397–401 (2015). https://doi.org/10.1007/s00454-014-9655-9
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DOI: https://doi.org/10.1007/s00454-014-9655-9