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arXiv · 1710.09672

1-skeletons of the spanning tree problems with additional constraints

Vladimir Bondarenko·Andrei Nikolaev·Dzhambolet Shovgenov
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Abstract

We consider the polyhedral properties of two spanning tree problems with additional constraints. In the first problem, it is required to find a tree with a minimum sum of edge weights among all spanning trees with the number of leaves less or equal a given value. In the second problem, an additional constraint is the assumption that the degree of all vertices of the spanning tree does not exceed a given value. The decision versions of both problems are NP-complete. We consider the polytopes of these problems and their 1-skeletons. We prove that in both cases it is a NP-complete problem to determine whether the vertices of 1-skeleton are adjacent. Although it is possible to obtain a superpolynomial lower bounds on the clique numbers of these graphs. These values characterize the time complexity in a broad class of algorithms based on linear comparisons. The results indicate a fundamental difference in combinatorial and geometric properties between the considered problems and the classical minimum spanning tree problem.

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Vladimir Bondarenko, Andrei Nikolaev, Dzhambolet Shovgenov. 2017-10-26. 1-skeletons of the spanning tree problems with additional constraints. https://doi.org/10.3103/s0146411617070033

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