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We consider several families of combinatorial polytopes associated with thefollowing 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 thefollowing method. We say that a family $P$ is affinely reduced to a family $Q$if for every polytope $p\in P$ there exists $q\in Q$ such that $p$ is affinelyequivalent to $q$ or to a face of $q$, where $\dim q = O((\dim p)^k)$ for someconstant $k$. Under this comparison the above-mentioned families are splittedinto two equivalence classes. We show also that these two classes are simpler(in the above sence) than the families of poytopes of the following problems:set covering, traveling salesman, 0-1 knapsack problem, 3-satisfiability, cubicsubgraph, partial ordering. In particular, Boolean quadratic polytopes appearas faces of polytopes in every of the mentioned families.

A special role of Boolean quadratic polytopes among other combinatorial polytopes