On affine reducibility of combinatorial polytopes

Abstract

283This paper considers families of combinatorial polytopes for which the problem of recognizing the nonadjacency of two arbitrary vertices is NP com plete. It is shown that all such polytopes contain spe cial 0/1 programming problems as faces, for which the problem of recognizing the nonadjacency of verti ces is NP complete too.A combinatorial polytope is usually specified as fol lows. Consider a set X ⊆ {0, 1} d of vertices of the d dimensional 0/1 cube satisfying certain constraints, specific for each problem. The polytope itself is defined as the convex hull of the set X, which is simul taneously the vertex set of the polytope. In what fol lows, we identify polytopes with their vertex sets.The study of graph properties of combinatorial polytopes became particularly popular in the 1970s [1,2] in relation to the potential possibility of solving the corresponding combinatorial problems by using the local search technique (which, in particular, uses the simplex method). At present, it has become a rule to begin searching for an efficient solution of a combina torial problem with studying properties of the associ ated polytope. One of the popular problems in this direction is searching for a simple adjacency criterion for polytope vertices. It is known from papers by Papadimitriou [3], Matsui [4], Bondarenko and Yurov [5], Alfakih and Murty [6], and Fiorini [7] that, for some families of combinatorial polytopes, the problem of recognizing the nonadjacency of vertices is NP complete. This gives rise to the natural question of whether the polytopes considered in [3-7] have a common constructive special feature, which deter mines the similarity of their properties. It turns out (details are given below) that each of these polytopes contains, as a face, the polytope corresponding to a 0/1 programming problem of special form. One of the properties of the latter polytope is the NP complete ness of the recognition of vertex nonadjacency.We proceed to list the polytopes considered in [3][4][5][6][7]. We conventionally divide them into two groups. The polytopes from the first group can be well described as 0/1 programming problems. In this case, we set x = (x 1 , x 2 , …, x d ) ∈ {0, 1} d . The second group consists of polytopes related to problems on graphs. If the graph corresponding to a problem under consideration is the complete unoriented graph G = (V, E) on n vertices, then the value of a coordinate x ij with 1 ≤ i < j ≤ n of a vector x ∈ {0, 1 symbolizes the presence or absence of the edge (v i , v j ) in the subgraph G(x) of the graph G. In such cases, x is often referred to as the characteristic vector of the graph G(x). Similarly, for the complete directed graph D = {V, A} on n vertices, the set of vectors x ∈ {0, 1} n(n -1) is in one to one cor respondence with the set of all subgraphs D(x) in the graph D.The polytope of covers SC md is defined as the con vex hull of all solutions x ∈ {0, 1} d of the inequality Ax ≥ 1, where A is an m × d, а 1 -m 0/1 matrix and 1 is the m vector all of whose coordinates equal 1...

A Special Role of Boolean Quadratic Polytopes among Other Combinatorial Polytopes

“…That is 3 ∝ 3AP in the sense of relation ≤ . However, from inequality (17), theorem 9, theorem 13 and DCP ∝ 3 [21] it follows that the reduction 3 ∝ 3AP is impossible. Now we show, that SSP ∝ 3AP.…”

“…Relying on definitions 1 and 3, we can introduce an analogue of Cook-Karp-Levin polynomial reducibility [14] for families of polytopes (as it was done in [21]).…”

“…Previously in [21,24], there have been found the following relations for several families of combinatorial polytopes with the property of NP-completeness of nonadjacency relation.…”

“…Therefore, if every -dimensional 0/1-polytope is a face of ATSP , then is superpolynomial in . In [21,24] it was shown that so-called double covering polytopes are affinely reduced to knapsack polytopes, set covering polytopes, cubic subgraph polytopes, 3-SAT polytopes, partial order polytopes, and traveling salesman polytopes. The linear optimization on double covering polytopes is NP-hard and the problem of checking nonadjacency on these polytopes is NP-complete [25].…”

A special role of Boolean quadratic polytopes among other combinatorial polytopes

“…In [22,26] it was shown that so-called double covering polytopes are affinely reduced to knapsack polytopes, set covering polytopes, cubic subgraph polytopes, 3-SAT polytopes, partial order polytopes, and traveling salesman polytopes. The linear optimization on double covering polytopes is NP-hard and the problem of checking nonadjacency on these polytopes is NP-complete [27].…”