Volume 23, No 3, 2016, P. 61-80

UDC 519.854
A. N. Maksimenko
Complexity of combinatorial optimization problems in terms of face lattice of associated polytopes

Abstract:
This paper deals with the following question: Can combinatorial properties of polytopes help in finding an estimate for the complexity of the corresponding optimization problem? Sometimes, these key characteristics of complexity were the number of hyperfaces of the polytope, diameter and clique number of the graph of the polytope, the rectangle covering number of the vertex-facet incidence matrix, and some other characteristics. In this paper, we provide several families of polytopes for which the above-mentioned characteristics differ significantly from the real computational complexity of the corresponding optimization problems. In particular, we give two examples of discrete optimization problem whose polytopes are combinatorially equivalent and they have the same lengths of the binary representation of the coordinates of the polytope vertices. Nevertheless, the first problem is solvable in polynomial time, while the second problem has exponential complexity.
Ill. 1, bibliogr. 22.

Keywords: NP-complex problem, vertex-facet incidence matrix, combinatorial equivalence, graph of a polytope, graph clique number, extended formulation, cyclic polytope.

DOI: 10.17377/daio.2016.23.515

Alexander N. Maksimenko ¹
1. Yaroslavl State University,
14 Sovetskaya St., 150000 Yaroslavl, Russia
e-mail: maximenko.a.n@gmail.com

Received 13 October 2015
Revised 19 April 2016

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