1-Skeletons of the Spanning Tree Problems with Additional Constraints
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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.
Keywordsspanning tree 1-skeleton clique number NP-complete problem Hamiltonian path
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- 1.Belov, Y.A., On clique number of matroid skeleton, in Modeli issledovaniya operacii v vychislitel’nykh sistemakh (Models of Operations Research in Computer Systems), Yaroslavl, 1985, pp. 95–100Google Scholar
- 2.Bondarenko, V.A., Complexity bounds for combinatorial optimization problems in one class of algorithms, Russ. Acad. Sci. Dokl. Math., 1993, vol. 328, no 1, pp. 22–24.Google Scholar
- 3.Bondarenko, V.A. and Maksimenko, A.N., Geometricheskie konstruktsii i slozhnost’ v kombinatornoi optimizatsii (Geometric Constructs and Complexity in Combinatorial Optimization), Moscow: LKI, 2008.Google Scholar
- 9.Goemans, M.X., Minimum bounded-degree spanning trees, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, 2006, pp. 273–282Google Scholar
- 15.Singh, M. and Lau, L.C., Approximating minimum bounded degree spanning trees to within one of optimal, Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, 2007, pp. 661–670Google Scholar