A special role of Boolean quadratic polytopes among other combinatorial polytopes

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 $P$ is affinely reduced to a family $Q$ if for every polytope $p\in P$ there exists $q\in Q$ such that $p$ is affinely equivalent to $q$ or to a face of $q$, where $\dim q = O((\dim p)^k)$ for some constant $k$. 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 sence) than the families of poytopes 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 of the mentioned families.


Introduction
In 1954, Dantzig, Fulkerson, and Johnson [7] solved a 49-city traveling salesman problem via considering a polytope of this problem. This idea turned out quite fruitful. Since then, there were publicated hundreds of papers about properties of various combinatorial polytopes. In particular, a lot of attention was paid to properties of graphs (1-skeletons) of polytopes (such as criterion of adjacency, diameter, clique number) and complexities of extended formulations.
In this paper we compare combinatorial characteristics of complexity for several families of such polytopes. The main results was anonced in 2011 [20] and in 2012 [22], but only recently the full version has been translated into English.
It is natural to consider the following method for comparing of polytopes. If a polytope p is affinely equivalent to a (not necessary proper) face of a polytope q, then p can not be more complicated than q in any reasonable sense. Below this fact is denoted by p ≤ A q. If we can compare two polytopes in this sense, then we can compare their characteristics of complexity. For example, if p ≤ A q then the graph of p is a subgraph of q, moreover, the face lattice of p is embedded into the face lattice of q.
We note that in recent times the most widespread is a little bit different method for comparing polytopes. This is related with the notion of an extended formulation of a polytope (see for example [6] and [16]). A polytope q is called an extension (or an extended formulation) of a polytope p if there exists an affine map π with Supported by the project No. 477 of P.G. Demidov Yaroslavl State University within State Assignment for Research. π(q) = p. The extension complexity of a polytope p is the size (i.e. number of facets) of its smallest extension. We will denote by p ≤ E q the fact that a (not necessary proper) face of a polytope q is an extension of p. It is clear, that For example, Yannakakis [33] showed that the matching polytopes and the vertex packing polytopes are not more complicated (in the sense of ≤ E ) than the traveling salesman polytopes. In [19] it was shown that polytopes of any linear combinatorial optimization problem (among them are cut polytopes, 0-1 knapsack polytopes, 3-satisfiability polytopes and many other combinatorial polytopes; see [23] for precise definition in English) are not more complicated than the traveling salesman polytopes. One year later in [23] 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. In general, it seems that this statement is true for the family of polytopes of any known NP-hard problem. (The results of this paper are another confirmation.) That is families of polytopes of NP-hard problems are not distinguishable while comparing by ≤ E . On the one hand, this is convenient for obtaining the results of a general nature. For example, since the extension complexity of cut polytopes is superpolynomial [12], then the same is true for (almost) all other families of combinatorial polytopes. On the other hand, it is well known that these families are significantly different from each other. For example, any two vertices of the cut polytope constitute an edge (1-face) of this polytope (i.e. the graph of this polytope is complete) [28]. Whereas the checking of nonadjacency of vertices of traveling salesman polytopes is NP-complete [29]. So, it is useful to have a more sensitive method of comparing, like the mentioned above ≤ A .
Earlier in [21,25] it was shown that knapsack polytopes, set covering polytopes, cubic subgraph polytopes, 3-SAT polytopes, partial order polytopes, and traveling salesman polytopes contain so-called double covering polytopes as faces. The linear optimization on double covering polytopes is NP-hard and the problem of checking nonadjacency on these polytopes is NP-complete [26]. Consequently, the same is true for the mentioned families.
In this paper we show, that (in the sense of relation ≤ A ) Boolean quadratic polytopes (and cut polytopes), quadratic linear ordering polytopes, and quadratic assignment polytopes lie in one equivalence class. Set partition polytopes, set packing polytopes, stable set polytopes, and 3-assignment polytopes lie in another (more complicated) equivalence class and they are simpler (w.r.t. ≤ A ) then double covering polytopes. Thus, 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. Naturally, the following question arises. Whether this family is "the purest" one? The answer to this question was provided in [24]. One can construct infinitely many families of Boolean p-power polytopes (Boolean quadratic polytopes is called Boolean 2-power polytopes) such that each (p + 1)-family is more pure than p-family for p ∈ N, p ≥ 2.
The rest of the paper is organized as follows. Section 2 provides a definition of affine reducibility and its properties. As an example we show that Boolean quadratic polytopes (BQP) are affinely reduced to stable set polytopes (SSP), but SSP can not be affinely reduced to BQP. In section 3 it is shown that SSP are equivalent (in the sense of affine reduction) to set packing polytopes and set partition polytopes. In section 4 we consider double covering polytopes (DCP) and prove that SSP ∝ A DCP, but DCP ∝ A SSP. In section 5 it is shown that SSP are equivalent to 3-assignment polytopes. In section 6 we consider quadratic linear ordering polytopes and quadratic assignment polytopes and show that they are equivalent to BQP.

Affine reducibility and Boolean quadratic polytopes
Let's consider the following partial order on the set of all convex polytopes. Definition 1. The fact that a polytope p is affinely equivalent to a polytope q or to a face of q will be denoted by p ≤ A q. If p is affinely equivalent to q itself we will use designation p = A q.
This relation is useful for estimation of combinatorial characteristics of polytopes. For example, if p ≤ A q then the number of i-faces of p is not greater than the number of i-faces of q for 0 ≤ i ≤ dim p. It is clear also that the number of facets of p is not greater than the number of facets of q. Furthermore, the graph (1-skeleton) of p is a subgraph of the graph of q. Hence we can compare clique numbers of these graphs and the like. Here it should be noted that the clique number is the lower bound of complexity of the appropriate linear optimization problem in the class of so-called direct type algorithms [3]. In the case p ≤ A q we can also compare the extension complexities of these polytopes [12]. The same is true for the rectangle covering bound [12].
On the other hand, this relation is useless, for example, for estimating diameters of graphs of polytopes. But the diameter of graph can hardly be seen as a characteristic of complexity. It is not greater than 2 for TSPpolytopes [27] and it is equal to 1 for Boolean quadratic polytopes [28]. This does not correspond to the real complexity of these problems.
It turns out that this relation allows to form up the currently known families of combinatorial polytopes in hierarchical order. At the very bottom of this hierarchy there are Boolean quadratic polytopes and cut polytopes.
Boolean quadratic polytope is the convex hull of the set It should be noted that the notation BQP n is commonly used for the convex hull, but in order to brevity we do not make distinctions between polytopes themselves and the sets of its vertices. The same remark applies to all other polytopes discussed below. This polytope is also known as a correlation polytope [9] and a clique polytope [3]. Moreover BQP n is directly related by so-called covariant mapping with cut polytope, usually denoted by CUT n [8]. Using the notation of definition 1 this relationship may be written as The proof of this property is simple. Note that such relations are normal for families of combinatorial polytopes.
In order to illustrate the basic ideas of this paper, we consider another family of polytopes, which is closely related with BQP n .
Let G = (V, E) is undirected graph with the set of vertices V = {v 1 , v 2 , . . . , v k } and the set of edges E. To each vertex v i , 1 ≤ i ≤ k, we associate the component y i of the vector y = (y 1 , . . . , y k ) ∈ R k . The stable set polytope of a graph G is the convex hull of the set This polytope also known as a vertex packing polytope. Futhermore, by affine map- it is related to the vertex covering polytope of a graph G: Let us note that BQP n is uniquely determined for a fixed n, whereas the notation SSP k hides a set of k-dimensional polytopes. For example, if a graph G has no edges then SSP k is a cube. If G is a complete graph then SSP k is a simplex. Hereinafter the SSP k will be associated with "the most complicated" polytope in this set. More precisely, for a polytope p and for fixed k inequality Generalizing this agreement we obtain Definition 3. Let P and Q are sets of polytopes. Then the record indicates that for every p ∈ P there exists q ∈ Q such that p ≤ A q.
This agreement allows us to deduce an analogue of the property 2 for SSP k .
Furthermore, using this notation BQP n and SSP k can be compared.
(A similar result is given in [12], but they used relation ≤ E and k = 2n 2 .) Proof. The equality subject to x ij ∈ {0, 1}. It remains to transform each of them in an inequality of the form y i + y j ≤ 1 from (2). For this we introduce n(n + 1) new variables: Then the restrictions (3) are equivalent to subject to integrality of all variables. Obviously, the last two equalities (more precisely, n(n + 1)/2 equalities) define some face of a polytope SSP k , where k = n(n + 1), defined by the system of n(2n − 1) inequalities Moreover, the equalities (4) connect this face with the polytope BQP n by nondegenerate affine mapping.
Remark 6. For k ≥ 2 relation SSP k ≤ A BQP n is not satisfied for any n. Since BQP n is a 2-neighborly polytope [28].
Relying on definitions 1 and 3, we can introduce an analogue of Cook-Karp-Levin polynomial reducibility [13] for families of polytopes (as it was done in [21]).

Definition 7.
A family of polytopes P is affinely reduced to a family Q if there exists r ∈ N such that for every polytope p ∈ P there is q ∈ Q with p ≤ A q and dim q = O ((dim p) r ). Designation: P ∝ A Q.
In such terminology the theorem 5 and the remark 6 take the following form: We list some obvious properties of this kind of reduction.
Suppose that the most complex polytopes in P have some of the following properties: 1) superpolynomial (in the dimension of a polytope) number of vertices and facets, 2) superpolynomial clique number of the graph of a polytope, 3) NP-completness of nonadjacency relation, 4) superpolynomial extension complexity, 5) superpolynomial rectangle covering bound. Then the most complex polytopes in Q have the same properties.

Set packing and set partition polytopes
Let G = {g 1 , . . . , g n } be a finite set and S = {S 1 , . . . , S d } ⊆ 2 G be a set of subsets of G. Consider a subset T ⊆ S. If every g i ∈ G belongs no more (no less) than one of elements of T then T is called a packing (covering) of the set G. Covering, which is both the packing, called a partition of the set G.
Let A = a ij be n × d matrix of incidences of elements of G and elements of S: For every subset T ⊆ S we consider its characteristic vector x = (x j ) ∈ R d : Denote the set of all such characteristic vectors by PACK d = PACK(S). It is evident that where 1 is the n-dimensional all 1 vector. The convex hull of PACK d is called the set packing polytope. Partition polytopes are defined similarly. The set of vertices PART d of the set partition polytope satisfies the equality It is clear that a set partition polytope is a face of a set packing polytope: Note that a stable set polytope is a special case of a set packing polytope: It is not difficult to prove that the families SSP = {SSP k }, PACK = {PACK d } and PART = {PART d } are equivalent in terms of affine reducibility.
Proof. Show that PACK d is a special case of SSP k for k = d. It is sufficient to note that inequality is equivalent to the set of inequalities The reduction PART ∝ A PACK is evident (see (6)). Now we show that SSP ∝ A PART. Consider auxiliary variables u ij = 1 − y i − y j , u ij ∈ {0, 1}. Then the inequalities y i + y j ≤ 1 in (2) can be replaced by equalities Here E is the set of edges in (2).

Double covering polytopes
The name "double covering polytopes" was used in [21] for a family of polytopes considered in [26].
Definition 10. The double covering polytope is the convex hull of the set where B ∈ R m×d is a 0-1 matrix, 2 is the m-dimensional all 2 vector, and each row of B contains exactly four 1's.
Previously in [21,25], there have been found the following relations for several families of combinatorial polytopes with the property of NP-completeness of nonadjacency relation.
Theorem 11 ([21, 25]). The family of double covering polytopes is affinely reduced to families of polytopes associated with the following problems: travelling salesman, knapsack, set covering, 3-satisfiability, cubic subgraph, partial ordering. Now we prove that stable set polytopes are simpler than double covering polytopes.
where |E| is the number of edges (inequalities) in the equation (2).
Proof. Let us look at the equation (2). For every edge {v i , v j } ∈ E we consider auxiliary variable u ij = 1 − y i − y j , u ij ∈ {0, 1}. Thus every inequality y i + y j ≤ 1 in (2) can be replaced by equality (7) y i + y j + u ij = 1.
Let u 0 be yet another auxiliary variable and let u 0 = 1. Hence the equality (7) is equivalent to According to the definition 10, a system of such equalities together with the requirement of integer variables defines the vertices of a double covering polytope DCP d for d = k + |E| + 1. The constraint u 0 = 1 defines a face of this polytope. Moreover this face is affinely equivalent to the given SSP k .
We now show that the affine reducibility in the opposite direction is not possible. Note that the NP-completness of adjacency relation is inherited when affine reducing (theorem 8). The family of double covering polytopes has this property [26], whereas for a stable set polytope the checking of adjacency is polynomial [5]. Hence, if P = N P then DCP can not be affinely reduced to SSP. It turns out that the latter is true even without the assumption P = N P . They are splitted into three pairs with the following property Assume that P is affine equivalent to some face conv{y 1 , . . . , y 6 } of SSP k . It is clear that the vertices y 1 , y 2 , . . . , y 6 inherit the property (8): (9) y 1 + y 2 = y 3 + y 4 = y 5 + y 6 .
We now show that there are two more vertices y 7 and y 8 of SSP k with (10) y 7 + y 8 = y 1 + y 2 .
Therefore we shall consider only those coordinates which values are different for every pair of vertices: All six sets are distinct, otherwise there would be identical vertices among y 1 , y 2 , . . . , y 6 . Under this condition, the two sets differ from each of the above six. Now we can define the points y 7 and y 8 : This 0-1 points differ from y 1 , y 2 , . . . , y 6 and equality (10) is satisfied for them. It remains to prove that y 7 and y 8 belong to the SSP k . That is if y i + y j ≤ 1 holds for y 1 , y 2 , y 3 , y 4 , y 5 , y 6 then it holds for y 7 and y 8 also. By equation (11), this condition is satisfied for i, j ∈ I. This is true for i ∈ I and j ∈ J also, since max(y 1 i + y 1 j , y 2 i + y 2 j ) = y 1 i + 1 = y 7 i + 1 = max(y 7 i + y 7 j , y 8 i + y 8 j ). It remains to check the condition for i, j ∈ J. If i ∈ S and j ∈S then y 7 i + y 7 j = y 8 i + y 8 j = 1 and the condition is fulfilled. Consider the case i, j ∈ S. If i, j ∈ U then y 1 i = y 1 j = 1. Hence y i + y j ≤ 1 is violated by y 1 for i, j ∈ U . The same is true if i and j both belong to one of the setsŪ , V ,V , W ,W . But for any i and j in S the latter is true. For example, if i ∈ U ∩ V ∩ W and j ∈Ū ∩V ∩ W then i, j ∈ W and so on. Hence for any i, j ∈ S the inequality y i + y j ≤ 1 is violated for at least one of the vertices y 1 , . . . , y 6 .
The same is true for the case i, j ∈S by symmetry.

Three index assignment polytopes
Consider a ground set S, |S| = m. Coordinates of a vector x ∈ R S×S×S we denote by x(s, t, u), where s, t, u ∈ S. Three index assignment (or 3-dimensional matching) problem can be formulated as the following 0-1 programming problem: x(s, t, u) = 1 ∀u ∈ S, (12) s∈S u∈S x(s, t, u) = 1 ∀t ∈ S, (13) t∈S u∈S x(s, t, u) = 1 ∀s ∈ S, (14) x(s, t, u) ∈ {0, 1} ∀s, t, u ∈ S, (15) where c(s, t, u) ∈ R is an input vector. By 3AP m denote the set of all vectors x ∈ R S×S×S satisfying restrictions (12)- (15). The convex hull of 3AP m is called the (axial) three index assignment polytope.
The first results about this polytope can be found in [10] and [2]. A more recent survey can be found in [30]. In the Russian-language papers there are given the lower bound for the clique number of the graph of 3AP m [3] and various properties of noninteger vertices of relaxations of this polytope (see for example [18]).
It is obvious that 3AP m is a special case of PART d : That is the family of three index assignment polytopes is affinely reduced to set partition polytopes: 3AP ∝ A PART. Using a standard reduction [13] from 3SAT to 3-dimensional matching, Avis and Tiwary [1] showed that 3SAT polytope is a projection of a face of a three index assignment polytope. That is 3SAT ∝ E 3AP in the sense of relation ≤ E . However, from inequality (16), theorem 9, theorem 13 and DCP ∝ A 3SAT [21] it follows that the reduction 3SAT ∝ A 3AP is impossible. Now we show, that SSP ∝ A 3AP. Therefore 3AP lies in one equivalence class (in the sense of ∝ A ) with SSP, PART, and PACK.
For the graph G(V, E) in the equation (2) we denote by the set of isolated vertices.
Proof. The ground set S for the 3AP m will consist of three types of elements: (1) v andv for every isolated vertex v ∈ W .
(2) e for every edge e ∈ E.
(3) (e, v) for every e ∈ E and v ∈ e. Now we construct the set of triples Q ⊂ S × S × S such that the face is affinely equivalent to the conv(SSP k ). For every v ∈ W the set Q contains four triples: There are no other triples containing v orv. Hence, if x ∈ F then for every v ∈ W we have only two cases: (1) e, e, e for every e ∈ E.
(3) e, e, (e, v) for every e ∈ E and v ∈ e. (4) (e iq , v), (e iq+1 , v), e iq for every nonisolated v and for every e iq ∈ E(v), where addition q + 1 is performed modulo p = d G (v). We list some properties of the vertices of the face F . Note that for every (e, v) ∈ T the set Q contains exactly two triples with (e, v) in the third entry: (e, v), (e, v), (e, v) and e, e, (e, v) . Hence, the equation (12) for u = (e, v) is converted into x (e, v), (e, v), (e, v) + x e, e, (e, v) = 1.
Note also that for every e ∈ T the set Q contains exactly three triples with e in the first entry: e, e, e , e, e, (e, v 1 ) , and e, e, (e, v 2 ) , where e = {v 1 , v 2 }. Hence, the equation (14) for s = e is converted into x e, e, e + x e, e, (e, v 1 ) + x e, e, (e, v 2 ) = 1.
Reasoning by analogy, we obtain the following equation for every nonisolated v and for every e iq ∈ E(v), where addition q + 1 is performed modulo p = d G (v). Hence, Moreover, since That is x e, e, (e, v) = x e ′ , e ′ , (e ′ , v) for any two edges e and e ′ , v ∈ e, v ∈ e ′ . It is not difficult to see that for the vertices of the face F all variables x(s, t, u) are expressed linearly in terms of x e, e, (e, v) and an inequality (17) is an inequality y i + y j ≤ 1 in (2).

Quadratic linear ordering polytopes and quadratic assignment polytopes
We begin by describing the linear ordering problem in terms of graph theory. Let D = (V, A) be a digraph, where V = {1, 2, . . . , m} is a vertex set. We assume that D is complete. That is (i, j) ∈ A and (j, i) ∈ A for any i, j ∈ V , i = j. An acyclic tournament 1 in digraph D is called a linear ordering.
Consider a characteristic vector y ∈ R m(m−1)/2 for a linear ordering L in D. The coordinates y ij , 1 ≤ i < j ≤ m, of y are Denote by LOP m the set of characteristic vectors of all linear orderings in D.
The convex hull of LOP m is called the linear ordering polytope. LOP m can also be defined as the set of integer solutions y ∈ {0, 1} m(m−1)/2 of the 3-dicycle inequalities (see for example [14]): In [4] the quadratic linear ordering polytope is defined as follows. Let where (i, j) ≺ (k, l) means that either i < k or i = k and j < l. For every pair of distinct variables y ij and y kl there is introduced a new variable (19) z ijkl = y ij y kl , (i, j, k, l) ∈ I.
Denote by QLOP m the set of all vectors z ∈ {0, 1} d , d = m 2 m 2 + 1 /2, with coordinates y ij and z ijkl satisfying conditions (18) and (19). The convex hull of QLOP m is called the quadratic linear ordering polytope.
We show that an affine reduction in the opposite direction is also possible.
Proof. The idea of the proof is simple. LOP m contains an n-dimensional cube as a proper face. In the transformation LOP m to QLOP m this cube turns into a Boolean quadratic polytope. Note that equalities y ij = 0 and y ij = 1 defines supported hyperplanes for LOP m and for QLOP m . Let We set (20) Only variables y ij are unfixed where i is odd and j = i + 1. Let us check 3-dicycle inequalities (18). Suppose i < j < k, we have two cases: (1) If (i, j) / ∈ J then y ij = y ik = 0. Thus the inequality (18) is transformed into 0 ≤ y jk ≤ 1.
(2) If (i, j) ∈ J then i is odd, j = i + 1 is even, and k > i + 1. Hence, the inequality (18) is equivalent to 0 ≤ y ij ≤ 1. Therefore n variables y i i+1 , where i is odd, may take the values 0 or 1 independently of each other. Consequently, hyperplanes (20) define a face of LOP m and this face is an n-dimensional cube.
Look at the variables z ijkl , (i, j, k, l) ∈ I. If (i, j) / ∈ J or (k, l) / ∈ J then z ijkl = 0. In the case (i, j) / ∈ J and (k, l) / ∈ J we have z ijkl = y ij y kl , and besides y ij and y kl are free variables.
Thus there is the following nondegenerate affine map between the face of QLOP m and BQP n : The story for quadratic assignment polytopes is repeated almost exactly. The set of vertices 2AP m of the assignment polytope (or Birkhoff polytope) consists of vectors y ∈ {0, 1} m×m satisfying the conditions Define new variable z ijkl like (19): Denote by QAP m the set of all vectors z ∈ {0, 1} d , d = m 2 + m 2 2 , with coordinates y ij and z ijkl satisfying conditions (21), (22), and (23). The convex hull of QAP m is called the quadratic assignment polytope. It is also useful to consider the quadratic semi-assignment polytope [32]. In the definition of its vertex set QSAP m the condition (22) is omitted. This connection is used in [15] for obtaining valid inequalities for QAP m . In particular, QAP m is a 2-neighborly polytope (every two vertices constitute an edge of it), since BQP n is 2-neighborly. In [15] it is also shown that the linear ordering polytope LOP m and the traveling salesman polytope TSP m are projectons of QAP m : Note that the affine reductions LOP ∝ A QAP and TSP ∝ A QAP are impossible, since LOP m is not 2-neighborly for m ≥ 3 [34] and TSP m is not 2-neighborly for m ≥ 6 [27].
Proof. By analogy with the proof of theorem 17 it is sufficient to show that the Birkhoff polytope 2AP m has an n-dimensional cube as a face. Let Then the equalities y ij = 0 for every (i, j) / ∈ J define the required face.

Resume
Boolean quadratic polytopes, cut polytopes, quadratic linear ordering polytopes, and quadratic assignment polytopes are in one class of equivalence within the framework of affine reducibility. A bit more complicated class contains stable set polytopes, set packing polytopes, set partitioning polytopes, and n-index assignment polytopes for n ≥ 3. An even more complicated are double covering polytopes, 3satisfiability polytopes, set covering polytopes, knapsack polytopes, cubic subgraph polytopes, partial ordering polytopes, traveling salesman polytopes. The problem of partitioning of these families into equivalence classes is not solved completely. Nevertheless, Fiorini [11] proved that k-satisfiability polytopes and m-satisfiability polytopes are in different classes for k = m. Moreover all of them are simpler than traveling salesman polytopes. From the other hand, the families of so-called Boolean p-power polytopes also are in different classes for distinct values of p [24]. Besides, Boolean p-power polytopes are simpler than Boolean quadratic polytopes (in the sense of affine reduction).
However if in the definition of affine reducibility (definition 7) we replace the relation ≤ A by relation ≤ E (recall that we write p ≤ E q if a face of a polytope q is an extension of p) then all the mentioned families of polytopes fall into one class of equivalence, since the polytope P of any linear combinatorial optimization problem is an affine image of a face of BQP n , where n is polynomial in the dimension of P [23].
Thus the affine reduction is a more delicate instrument (versus extending) for comparing the families of combinatorial polytopes. The most complicated (more precisely, the richest in its properties) is a family of traveling salesman polytopes. Families of Boolean p-power polytopes are more simple than any other of the above. More precisely they contain the minimum number of superfluous details (with respect to other families associated with NP-hard problems). Moreover, apparently, combinatorial and geometric properties determining NP-hardness reach the highest concentration precisely in Boolean p-power polytopes.
Proceed to a more precise formulation. Using the above results it is easy to derive the following relations.