A counterexample to a conjecture of Laurent and Poljak
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The metric polytope m(n) is the polyhedron associated with all semimetrics on
n nodes. 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<9 and, in particular, for the 1 550 825 600 vertices
of m(8). While the overwhelming majority of the known vertices of m(9) satisfy
the Laurent-Poljak conjecture, we exhibit a fractional vertex not adjacent to
any integral vertex.