Abstract
We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set, 3-assignment. For comparing two families of polytopes we use the following method. We say that a family
Highlights
In 1954, Dantzig, Fulkerson, and Johnson [8] solved a 49-city traveling salesman problem via considering a polytope of this problem
We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set, 3-assignment
A special role of Boolean quadratic polytopes among other combinatorial polytopes with the following example. It has been shown by Billera and Sarangarajan [3] that every 0/1-polytope p ⊂ Rd with k vertices is affinely equivalent to a face of the asymmetric traveling salesman polytope: p ≤A ATSPn for n ≥ (4(2d − k) + 1)d
Summary
In 1954, Dantzig, Fulkerson, and Johnson [8] solved a 49-city traveling salesman problem via considering a polytope of this problem. One year later in [22] it was shown that polytopes of any linear combinatorial optimization problem are not more complicated than the cut polytopes and the 0-1 knapsack polytopes It seems that this statement is true for the family of polytopes of any known NP-hard problem. A special role of Boolean quadratic polytopes among other combinatorial polytopes with the following example It has been shown by Billera and Sarangarajan [3] that every 0/1-polytope p ⊂ Rd with k vertices is affinely equivalent to a face of the asymmetric traveling salesman polytope:. The family of Boolean quadratic polytopes is more pure example of a family of polytopes associated with NP-hard problems They do not have extra details like NP-completness of adjacency relation and the like. 20/1-polytope is the convex hull of a subset of the vertices of the cube [0, 1]d
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