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A counterexample to the dominating set conjecture

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Abstract

The metric polytope met n is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities x ij  −  x ik  −  x jk  ≤  0 and x ij  +  x ik  +  x jk  ≤  2 for all triples i, j, k of {1,..., n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n  ≤  8 and, in particular, for the 1,550,825,600 vertices of met8. While the overwhelming majority of the known vertices of met9 satisfy the conjecture, we exhibit a fractional vertex not adjacent to any integral vertex.

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  1. Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, ON, Canada

    Antoine Deza & Gabriel Indik

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  1. Antoine Deza
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  2. Gabriel Indik
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Correspondence to Antoine Deza.

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Deza, A., Indik, G. A counterexample to the dominating set conjecture. Optimization Letters 1, 163–169 (2007). https://doi.org/10.1007/s11590-006-0001-x

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  • DOI: https://doi.org/10.1007/s11590-006-0001-x

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