A Special Role of Boolean Quadratic Polytopes among Other Combinatorial Polytopes

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 is affinely reduced to a family if for every polytope ∈ there exists ∈ such that is affinely equivalent to or to a face of , where dim = ((dim ) ) for some constant . Under this comparison the above-mentioned families are splitted into two equivalence classes. We show also that these two classes are simpler (in the above sense) than the families of polytopes of the following problems: set covering, traveling salesman, 0-1 knapsack problem, 3-satisfiability, cubic subgraph, partial ordering. In particular, Boolean quadratic polytopes appear as faces of polytopes in every mentioned families.Article is published in the author's wording.

On a family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation

“…Так как семейство P part аффинно сводится к P stab [8], достаточно показать, что для любого графа G = (V, E), |V | > 0, |E| > 0, можно построить матрицу A с тремя единицами в каждой строке, что P stab (G) A P part (A).…”

“…С другой стороны, известно [5][6][7], что критерий смежности вершин полиномиален для многогранников следующих NP-полных задач: задача о независимом множестве в графе, задачи об упаковке и разбиении множества, трехиндексная задача о назначениях. В [8] было показано, что каждый многогранник из этих четырех семейств является гранью многогранника двойных покрытий, но, при этом, многогранник двойных покрытий…”

On Vertices of the Simple Boolean Quadric Polytope Extension

“…In recent years, the Boolean quadric polytope has been under the close attention in connection with the problem of estimating the extension complexity [13]. An extension of the polytope P is another polytope Q such that P is the image of Q under a linear map.…”

“…The basis is the first four blocks as shown in Table 4. We can use the constraints (12)- (13) to establish the relationship between the coordinates:…”

“…All coordinates of the constructed point are either already directly expressed in terms of x, or can be found using the equations (11)- (13). Thus, it is a unique solution of the system (11)- (14) and a vertex of the polytope SAT P LP (n, n) with a denominator n + 1.…”