1. Introduction
For a positive integer , consider the set of matrices with rank and no repeated row or column. How many matrices in are maximal, in the sense that they are not a submatrix of another matrix in ? We show that has surprisingly few maximal elements up to permutations of rows and columns, see Theorem 19. This is achieved by parametrizing these matrices by graphs on nodes, whose edges have integer weights in .
1.1. Motivation
Matrices with entries are studied in communication complexity. This field is concerned with quantifying the amount of information that two or more parties have to exchange in order to collaboratively evaluate a function [11].
We consider the simplest communication model with two parties, that of deterministic protocols [16]. It is well known that the deterministic communication complexity of a Boolean function can be lower bounded by the base logarithm of the rank of its communication matrix :
A fundamental open problem known as the logrank conjecture [12] states that this simple bound is almost tight for all Boolean functions , in the sense that
Proving the logrank conjecture in the special case where is a maximal element of with is enough to imply the whole conjecture, because deterministic communication complexity is monotone under taking submatrices, and invariant under repeating rows or columns. In this sense, maximal elements of are the hardest instances for the logrank conjecture. To our knowledge, this fact has not been used in the communication complexity literature. Our hope is that the succinct representation of such maximal matrices as weighted graphs introduced in this paper can be used in a new approach to the logrank conjecture.
Matrices with entries also appear in discrete geometry and discrete optimization. Let be a polytope, where has rows denoted by , and has points denoted by , . The nonnegative matrix whose entry is the slack of the th vertex with respect to the th inequality , that is, , is referred to as a slack matrix of . A similar definition holds for polyhedral cones.
A polytope (or polyhedral cone ) is level provided that it admits a slack matrix with entries [6].
Some examples of level polytopes in the literature are Hanner polytopes [10], Birkhoff polytopes [1], or more generally, polytopes of the form for some totally unimodular^{1}^{1}1A matrix is said to be totally unimodular provided that the determinant of every square submatrix of is either , or , see for instance [13]. and integral, order polytopes [14], and stable set polytopes of perfect graphs [3].
It is an open problem to determine what function of describes the number of affine equivalence classes of level polytopes. In [2], it is conjectured that this number is at most . This conjecture is backed by experimental evidence: Bohn et al. [2] enumerate all affine equivalence classes of level polytopes for dimension . (We point out that the enumeration algorithm has been since then improved^{2}^{2}2The latest implementation of the code is due to Samuel Fiorini, Marco Macchia, Aurélien Ooms (Université libre de Bruxelles). The source code and the complete list of all slack matrices of nonisomorphic level polytopes up to dimension are available online at https://github.com/ulb/tl . and produced the complete database up to .)
Further related work is that of Grande and Rué [9], who provide a lower bound on the number of 2level matroid polytopes, for some constant . Finally, in [7], Gouveia et al.
completely classify polytopes with minimum positive semidefinite rank (which generalize
level polytopes) in dimension .We point out that if the logrank conjecture holds, every level polytope can be described as the projection of a polytope with at most facets. This implication follows from a classic result of Yannakakis [15] linking the extension complexity of a polytope to the nonnegative rank of its slack matrices. It is known that the extension complexity of the stable set polytopes of a perfect graph is at most [15], but whether or not the same proof strategy can be generalized to the entire class of level polytopes is still open.
1.2. Contribution and outline
We phrase the counting problem for matrices in terms of counting so called level configurations, that we formally define at the beginning of Section 2. Basically, a level configuration is a rank factorization of maximal matrix in . These configurations also capture (maximal) slack matrices of level cones and level polytopes.
In Section 2, we introduce the notion of linear equivalence for level configurations. Intuitively, two level configurations are linearly equivalent if they are two rank factorizations of the same matrix in . Moreover, we show that given some level polytope , we can associate it to a level configuration. Similarly, a level cone can be associated to a level configuration.
In Section 3, we present a lower bound of for the number of affine equivalence classes of level polytopes, that implies a lower bound for the number of linear equivalence classes of level configurations.
Moreover, in Section 4, we prove a first upper bound of for the number of linear equivalence classes of level configurations. Next, in Section 5 we present our first main result:
Theorem 1.
The total number of affine equivalence classes of level polytopes and the total number of linear equivalence classes of level cones is at most .
This theorem follows from upper bounding the number of faces of the correlation cone [4, Chapter 5].
Theorem 2.
The total number of linear equivalence classes of level configurations is at most .
2. Preliminary definitions and results
A (convex) polytope is said to be level if there is a finite system of linear inequalities^{3}^{3}3Throughout, we assume that systems of linear inequalities do not have repeated inequalities. with , and a finite point set such that
(1)  
and  (2)  
(3) 
where denotes the th row of and the th point of .
Let be a fulldimensional level polytope. We call a pair a maximal pair of level descriptions for if and satisfy (1) and (3) and are maximal with respect to these properties. Notice that the inequalities and are always part of if is a maximal pair of level descriptions for . Since we assume that is full dimensional, every level polytope admits a maximal pair of level descriptions. We call a matrix a maximal slack matrix of if for some maximal pair of level descriptions for , where has rows and has points.
Likewise, a (pointed, polyhedral) cone is said to be level if there is a finite system of homogeneous linear inequalities with
and a finite vector set
such that and for all and . The notions of maximal pair of level descriptions and maximal slack matrix are defined similarly for cones as for polytopes. As in the case of polytopes, we assume cones to be full dimensional. This guarantees the existence of a maximal pair of level descriptions and a maximal slack matrix for each level cone. Although it plays no role in this paper, we remark that maximal slack matrices are unique for level polytopes, but not necessarily for level cones.A level configuration is a pair , where and are two maximal sets of vectors linearly spanning such that the inner product is in for all and . Letting , we obtain a matrix that is the slack matrix of the level configuration . Two level configurations and are called linearly equivalent if there exists a nonsingular matrix such that
The next remark follows from the fact that, for a level configuration , the sets and are required to linearly span .
Remark 3.
For a level configuration , the slack matrix is a maximal element in .
Let us first observe that level polytopes and level cones can be interpreted as special instances of level configurations.
Lemma 4.
Consider a fulldimensional level cone , where , and is a maximal pair of level descriptions for . Then the pair
defines a level configuration.
Proof.
Since is fulldimensional, it is enough to prove that both and are maximal. This holds because the set of linear inequalities and the set are maximal with respect to the property , where is the th row of the matrix and is the th vector in for , . ∎
Lemma 5.
Consider a fulldimensional level polytope , where , , and is a maximal pair of level descriptions for . Then the pair
defines a level configuration.
Proof.
To show that is a level configuration, it is enough to show that both and are maximal. Since the system is maximal with respect to the property for all , , we obtain that is maximal. Also, as we noticed before, the inequality is present in .
In order to prove that is maximal, we have to show that there exists no with , , and , such that for all . Let us assume that there exists such a vector .
From we get that because . Since and , we have . Thus, for all implies that . In particular, is in the recession cone^{4}^{4}4Given a nonempty convex set , the recession cone of is the set of all directions along which we can move indefinitely and still be in , i.e. . of . Since is a polytope, its recession cone contains no vector besides . Hence, and , a contradiction. ∎
An alternative proof of Lemma 5 uses Lemma 4 and the fact that every level polytope naturally yields a level cone pointed at the origin. In fact, let be a level polytope , where , , and is a maximal pair of level descriptions for . Let for every and for every , . Then it is straightforward to check that is a level cone pointed at the origin, where and is the matrix whose rows are , . Moreover, is a maximal pair of level descriptions for .
We now show that level polytopes, cones, or configurations can be encoded using their maximal slack matrices, i.e. they can be encoded as some matrices. We show that this a valid encoding, i.e. given a maximal slack matrix we can reconstruct the original level polytope up to affine transformation, or reconstruct the level cone or
level configuration up to linear transformation. We would like to note that the analogous statement holds for general polytopes and general pointed cones and slack matrices, which are not necessarily maximal.
Lemma 6.
If two level configurations , admit the same slack matrix up to permutation of rows and columns, then and are linearly equivalent.
Proof.
Without loss of generality, we assume . Both and provide rank factorizations of the matrix with rank . Thus, there exists a nonsingular matrix such that
The claim follows. ∎
Corollary 7.
If two level polytopes admit the same maximal slack matrix up to permutation of rows and columns, then these polytopes are affinely equivalent. Similarly, if two level cones admit the same maximal slack matrix up to permutation of rows and columns, then these cones are linearly equivalent.
The next lemma shows that given a level configuration , we can find another level configuration with the same maximal slack matrix and where is a set of vectors.
Lemma 8.
Given a level configuration , there exists a level configuration such that and .
Proof.
Since linearly spans , there are vectors which linearly span . Let be the matrix whose th column equals for . Then the set
contains standard basis vectors .
Moreover, the level configuration , where
has the same maximal slack matrix as the level configuration .
Finally, since is a matrix and the set contains standard basis vectors , we have that is a set of vectors. ∎
Analogously one can show the following lemma:
Lemma 9.
Given a level configuration , there exists a level configuration such that and .
Lemma 8 can be strengthened for level configurations arising from level polytopes and level cones.
We recall the definition of simplicial core for polytopes, that appeared in [2, Definition 1], and generalize it to cones. A simplicial core for a polytope is a tuple of facets and vertices of such that each facet does not contain vertex but contains vertices , …, . A simplicial core for a cone is a tuple of facets and extreme rays of such that each facet does not contain extreme ray but contains extreme rays . The proof of next result follows from [2, Lemma 9 and Corollary 10]. For the sake of completeness, we present another version here.
Lemma 10.
Given a level polytope , there exists a level configuration such that , and is a maximal slack matrix of .
Proof.
Let define a maximal pair of level descriptions for , where and . By [8, Proposition 3.2], [2, Lemma 2], there exist facets and vertices such that is a simplicial core for . Let us assume that the facets are defined by the linear inequalities respectively. By Lemma 5, the pair of sets
defines a level configuration.
Define to be the matrix whose th row is given by . Due to the definition of simplicial core, the matrix is nonsingular. Now, let
Clearly, is a level configuration. Moreover, , since is a level configuration.
Let be the vectors in corresponding to the vertices of , respectively. Let be the matrix with th column equal to . Since is a simplicial core, is a nonsingular lowertriangular matrix, thus is unimodular.
Define
As before, is a level configuration. Since is a unimodular matrix and , we have that all vectors in are integral. Moreover, vectors are in , hence all vectors in are vectors. We conclude that is the desired level configuration. ∎
A result analogous to Lemma 10 holds for level cones. Its proof follows the proof of the previous lemma and is left to the reader.
Lemma 11.
Given a level cone , there exists a level configuration such that , and is a maximal slack matrix for .
3. Lower bound on the number of 2level polytopes
In this section we prove that the number of affine equivalence classes of level polytopes is . To do that we use a well known family of level polytopes: the family of stable set polytopes of bipartite graphs. First, we show that two nonisomorphic bipartite graphs lead to affinely nonequivalent stable set polytopes, whenever the minimum degree of both graphs is at least . Then, we use the result by [5] to show the lower bound for the number of isomorphism classes for bipartite graphs with minimum degree at least .
The stable set polytope of a graph , denoted by , is the convex hull of the characteristic vectors of stable sets of . Let be a bipartite graph with no isolated nodes, then the stable set polytope can be described in the following way:
(4) 
It is straightforward to verify that each of the above inequalities defines a facet of whenever is bipartite. Moreover, since is a fulldimensional polytope, different inequalities above define different facets of .
Claim 12.
Let be a node graph such that the minimum degree of a node in is at least . Then has a unique simple vertex , i.e. is the only vertex of contained in exactly facets of .
Proof.
Let us consider a vertex of , corresponding to a stable set . The vertex is contained in some facets defined by nonnegativity constraints and some facets defined by edge constraints. First, there are exactly facets corresponding to nonnegativity constraints, which contain the vertex . Second, to each edge incident to a node in corresponds a facet which contains . Since each node in has degree at least , there are at least
facets containing the vertex . Now, note that the vertex of corresponding to is contained in exactly facets, finishing the proof. ∎
Claim 13.
Given a node graph , the vertex is incident only to the vertices , of .
Proof.
The statement follows from the fact that the vertex of is incident only to the facets induced by nonnegativity constraints , . ∎
Lemma 14.
Let , be two graphs such that the minimum degree of nodes in and is at least . Then , are isomorphic if and only if , are affinely equivalent.
Proof.
Clearly, if and are isomorphic then , are affinely equivalent as well.
Now suppose that , are affinely equivalent, i.e. there exists a bijective affine map such that , where . By Claim 12, we have , and by Claim 13, we have for all , where is a bijection. Now, it is straightforward to verify that defines an isomorphism between and , because for a graph we have that if and only if , , define a triangular face of . ∎
Let be any set of cardinality . The result by [5, Lemma 7] shows that the number of labeled bipartite graphs with node set is at least and at most . Hence,
is an upper bound for the number of labeled bipartite graphs with node set and minimum degree less than . Thus there are at least
many labeled bipartite graphs with node set and minimum degree at least . Therefore, there are isomorphism classes of bipartite graphs on nodes with minimum degree at least . This fact together with Lemma 14 implies the following result.
Theorem 15.
The number of affine equivalence classes of level polytopes is .
By Lemma 5, every level polytope yields a level configuration. We obtain the following result:
Corollary 16.
The number of linear equivalence classes of level configurations is .
4. First upper bound for 2level configurations
The next theorem gives a first upper bound on the total number of maximal matrices with rank . This will be the basis for the refined upper bounds in Sections 5 and 6.
Theorem 17.
The number of maximal elements of (up to permutation of rows and columns) is .
Proof.
Let be the slack matrix for a level configuration . In our proof, we show that the matrix (up to permutation of rows and columns) can be reconstructed from a set of at most linear equations, where each linear equation belongs to a fixed collection of possible linear equations. This gives us the desired upper bound.
By Lemma 9, we can assume . From the maximality of , we get
Thus, knowing the linear space spanned by the vectors , , we are able to first reconstruct the set and then reconstruct the set .
The dimension of the linear space span of all the vectors , is . Since , the dimension of the linear span of the vectors , is at most . Thus, there is , , such that the linear spaces spanned by , and , are the same. Hence, to define the desired linear subspace, it is enough to select at most vectors . Each of these vectors is chosen in . The result follows. ∎
5. Second upper bound for 2level cones
In this section, we refine Theorem 17 for level configurations such that is a set of vectors and a set of integer vectors. Together with Lemmas 10 and 11, our refinement implies Theorem 1 from the introduction.
Theorem 18.
The total number of matrices (up to permutation of rows and columns) which are slack matrices of some level configuration with and is at most .
Proof.
The proof is analogous to the proof of Theorem 17. Due to maximality of , we have
Notice that for any , we have
Now if both and are integer vectors, we have and thus . Hence, for every the inequality defines a face of the cone
(5) 
We remark that, since , we have for all . Thus, the vector appears on the diagonal of the matrix . We deduce that the cone in (5) is linearly equivalent to the correlation cone [4, Chapter 5], which is defined as the conic hull of all matrices for .
The set is the set of all such that the vector lies in the following face of :
Note that is a pointed cone of dimension , hence each of its faces is uniquely defined by the sum of at most rays from the set of its extreme rays
Such a sum is always an integer vector in . Hence, the total number of possible faces of is at most . Moreover, since each face defines at most one possible , we have the desired upper bound on the total number of different sets , and hence on the total number of different level configurations , where and . ∎
6. Second upper bound for 2level configurations
In this section, we improve the upper bound in Theorem 17 for general level configurations, that is, without the extra assumptions of the previous section. This is our final result. It implies Theorem 2 from the introduction.
Theorem 19.
The number of maximal elements of (up to permutation of rows and columns) is .
Proof.
Let be the slack matrix of a level configuration . By Lemma 9, we may assume that and .
Let us show that there are vectors , , such that the lattice generated by equals the lattice generated by , …, . Let us start with a set of linearly independent vectors from . Now until , we iteratively replace by for some .
Note that if , then the lattice is a proper sublattice of for . Thus the determinant of equals the determinant of times an integer strictly larger than . Since , the determinant of the initial lattice is at most . Whenever , the determinant of decreases each time by an integer factor larger than or equal to . Thus, by the time , the cardinality of is at most .
Now we define maps and such that
For , we let . For , we let , where , are integer coefficients verifying . For , we let .
Thus
Let . Then . Now, similarly to the proof of Theorem 18, one can show that there are at most possible . The map is uniquely defined by the vectors . Thus, there are at most possible maps . Note that, by extending in the obvious way to a map defined on the whole ,
showing that there are at most , and so at most , possibilities for . ∎
7. Discussion
Similarly to Theorem 17, one can show that the total number of maximal matrices of rank with entries in (and no repeated row or column) is at most .
This gives an upper bound of on the number of linear equivalence classes of cones such that for all and , where has rows denoted by , and has vectors denoted by , .
Also, this gives an upper bound of on the number of affine equivalence classes of polytopes such that for all and , where has rows denoted by , and has points denoted by , .
We would like to point out that there are infinitely many affine equivalence classes of polytopes such that
already for and . This is due to the observation that every quadrilateral has a pair of outer and inner descriptions satisfying the above condition for . However, there are infinitely many affine equivalence classes of quadrilaterals.
We leave it as an open problem to fill the gap between the lower bound of and the upper bound of for the number of affinely inequivalent level polytopes and the number of linearly inequivalent level cones, and also find better estimates on the number of linear equivalence classes of level configurations.
Acknowledgments
We acknowledge support from ERC grant FOREFRONT (grant agreement no. 615640) funded by the European Research Council under the EU’s 7th Framework Programme (FP7/20072013).
This work was done while the authors were visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant
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