On the Linear Extension Complexity of Regular n-gons

On the Linear Extension Complexity of Regular -gons

Arnaud Vandaele Department of Mathematics and Operational Research, Faculté Polytechnique, Université de Mons, Rue de Houdain 9, 7000 Mons, Belgium. Emails: {arnaud.vandaele, nicolas.gillis}@umons.ac.be.    Nicolas Gillis    François Glineur Université catholique de Louvain, CORE and ICTEAM Institute, B-1348 Louvain-la-Neuve, Belgium; francois.glineur@uclouvain.be.
Abstract

In this paper, we propose new lower and upper bounds on the linear extension complexity of regular -gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size  of a polytope , and (ii) a rank- nonnegative factorization of a slack matrix of the polytope . The lower bound is based on an improved bound for the rectangle covering number (also known as the boolean rank) of the slack matrix of the -gons. The upper bound is a slight improvement of the result of Fiorini, Rothvoss and Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp. 658-668, 2012]. The difference with their result is twofold: (i) our proof uses a purely algebraic argument while Fiorini et al. used a geometric argument, and (ii) we improve the base case allowing us to reduce their upper bound by one when for some integer . We conjecture that this new upper bound is tight, which is suggested by numerical experiments for small . Moreover, this improved upper bound allows us to close the gap with the best known lower bound for certain regular -gons (namely, and ) hence allowing for the first time to determine their extension complexity.

Keywords. nonnegative rank, extension complexity, regular -gons, nonnegative factorization, boolean rank.

1 Introduction

An extended formulation (or extension) for a polytope is a higher dimensional polyhedron such that there exists a linear map with . The size of such an extended formulation is defined as the number of facets of the polyhedron . The size of the smallest possible extension of is called the (linear) extension complexity of and is denoted xc(). The quantity xc() is of great importance since it characterizes the minimum information necessary to represent . In particular, in combinatorial optimization, it characterizes the minimum size necessary to represent a problem as a linear programming problem (taking as the convex hull of the set of feasible solutions). Hence although might have exponentially many facets, might have only a few, providing a way to solve linear programs over much more effectively. An example of such a polytope is the permutahedron, that is, the convex hull of all permutations of the set with vertices and facet-defining inequalities, that can be represented as the projection of a polyhedron with facets [12].

The characterization of the extension complexity has attracted much interest recently; in particular lower bounds since they provide provable limits of linear programming to solve combinatorial optimization problems; see, e.g., [8]. For example, it was recently shown that the extension complexity of the matching polytope is exponential (in the number of vertices of the graph), answering a long-standing open question whether there exists a polynomial-size linear programming formulation for the matching problem [20] which implies that although it is solvable in polynomial time, the standard formulation cannot be written as a linear program with a polynomial number of inequalities.

Interestingly, most lower bounds for the extension complexity of polytopes are based on a well-known linear algebra concept: the nonnegative rank. The nonnegative rank of a nonnegative -by- matrix , denoted , is the minimum such that there exist a nonnegative -by- matrix and a nonnegative -by- matrix such that . The pair is a rank- nonnegative fatorization of . The link between the nonnegative rank and the extension complexity of a polytope, a seminal result of Yannakakis [24], goes as follows. Let be a polytope in dimension with

  • facets expressed as linear inequalities , and

  • vertices denoted .

The slack matrix of is defined as

Note that the slack matrix of a polytope is not unique since the inequalities can be scaled, and the rows and columns permuted but this does not influence its nonnegative rank; see [13] for more details. Note also that if is full dimensional. Then, we have

Moreover any nonnegative factorization of provides an explicit extended formulation for (with some redundant equalities):

where with for all , and . For example, the matrix

is a slack matrix of the regular hexagon (hence it has rank three) and has nonnegative rank equal to five:

This implies that the regular hexagon can be described as the projection of a higher dimensional polytope with 5 facets; see Figure 1 for an illustration.

6 facets

5 facets

Figure 1: Minimum-size extension of the regular hexagon.

In this paper, we focus on the extension complexity of regular -gons, and in particular on a new upper bound.

Extension complexity of regular -gons

In the remainder of this paper, we denote the slack matrix of the regular -gon (more precisely, any slack matrix; see Section 2 for a construction), hence equals the extension complexity of the regular -gon; see above. In the following, we describe several bounds for the nonnegative rank, focusing on the slack matrices of regular -gons.

Lower bounds. There exist several approaches to derive lower bounds for the nonnegative rank, which we classify in three classes:

  • Geometric. Using a counting argument and the facts that (i) any face of a polytope is the projection of a face of its extension, and (ii) any face is an intersection of facets, it can be shown that  [12]. Based on a refined geometric counting argument, Gillis and Glineur [10] described a stronger lower bound for the slack matrix of polygons111They actually derived this bound for linear Euclidean distance matrices, but it also applies to the slack matrix of polygons.: the nonnegative rank of must satisfy

    where the quantity is the maximal number of -faces of a polytope with vertices in dimension , attained by cyclic polytopes [18]; see also [25, p.257, Corollary 8.28]. We have

    where denotes a sum where only half of the last term is taken for if is even, and the whole last term is taken for if is odd. This bound can be generalized to any nonnegative matrix [10], but it becomes difficult to compute for non-slack matrices as it requires another quantity that is in general NP-hard to compute (namely, the restricted nonnegative rank, which is always equal to for the slack matrix of a polytope with vertices).

  • Combinatorial. These bounds are based on the sparsity pattern of the input matrix. The most well-known one is the rectangle covering bound (RCB) that counts the minimum number of rectangles necessary to cover all positive entries of the matrix, a rectangle being a subset of rows and columns for which the corresponding submatrix contains only positive entries; see [7] and the references therein. Note that the RCB is equal to the boolean rank; see, e.g., [4]. A closely related bound is the refined rectangle covering bound (RRCB) by Oelze, Vandaele, Weltge [19]: in addition to covering every positive entry by a rectangle, the RRCB requires that every 2-by-2 nonsingular submatrix is touched by at least two rectangles (note that the same rectangle can be used twice). For example, the RCB for the matrix

    is equal to two while the RRCB is equal to three. In fact, there are only three maximal rectangles (that is, rectangles not contained in any larger rectangle):

    and only two of them are required to cover all positive entries (the last two, which is the unique solution) while three are necessary to touching twice all rank-two positive submatrices (which is tight since this is a 3-by-4 matrix), e.g., the block touched only once with the RCB solution.

    Although these bounds can be rather strong in some cases, they are computationally very expensive, and only work well for matrices with ‘well located’ zero entries. For the slack matrices of the regular -gons, we could compute them up to (for larger , it would take several weeks of computation with our current formulation).

  • Convex Relaxations. Fawzi and Parrilo developed two lower bounds for the nonnegative rank based on a sum-of-squares approximation of the copositive cone [5, 6]. These bounds are very general as they can be computed for any nonnegative matrix; however they are typically weaker than the aforementioned lower bounds, in particular for slack matrices.

These bounds are compared for the regular -gons on

Figure 2: Comparison of lower and upper bounds for the nonnegative rank of the slack matrices of regular -gons, that is, . (Note that some bounds cannot be computed for all because of their high computational cost.)

Figure 2. We observe that the best lower bounds are the geometric bound from [10] and the rectangle covering bounds [7, 19] that coincide except for for which only the RRCB is tight (as it matches the best upper bound; see below).

Upper bounds. Ben-Tal and Nemirovski [3] gave an extension of the regular -gons when is a power of two ( for some ) with facets. They used this construction to approximate the circle with regular -gons which allowed them to approximate second-order cone programs with linear programs. This construction was slightly reduced to size in [11] (again, only for ). Kaibel and Pashkovich [15, 16] proposed a general construction for arbitrary of size . Fiorini, Rothvoss and Tiwary [9] improved the bound to , which is, to the best of our knowledge, the best known upper bound for regular -gons. These last bounds are based on a geometric argument using successive reflections to construct the regular -gon. Note that Shitov [21] proved an upper bound of for the nonnegative rank of any -by- rank-three nonnegative matrix, hence is applicable to the slack matrix of polygons.

As shown on Figure 2, prior to our new upper bound, the exact value of is not known for most values of larger than 9 as the best lower and upper bounds do not coincide. Therefore, the exact value of the extension complexity of many regular -gons is still unknown.

Table 1 also gives the best upper and lower bounds for up to 20.

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
RRCB [19] 5 6 6 7 7 7 7 8 ? ? ? ? ? ? ? ?
geometric [10] 5 6 6 6 7 7 7 7 7 8 8 8 8 8 8 9
Equation (1) 5 6 6 7 7 7 7 8 8 8 8 9 9 9 9 9
Table 1: Comparison of two lower bounds (first two rows) and the upper bound from Equation (1) for the nonnegative rank of regular -gons. Bold indicates the tight bounds, that is, bounds that coincide with the nonnegative rank.

Contribution of the Paper

In this paper, our contribution is mainly towfold. First, in Section 3, we derive an improved lower bound for the rectangle covering number of the slack matrix of regular -gons. We show that the following relation holds

which improves over the best known previous relation given by  [4]. Although this new lower bound does not improve the best known lower bounds for the nonnegative rank of the slack matrices of regular -gons (namely, the RRCB and the geometric bound; see previous paragraph), it is applicable to a broader class of matrices, namely those which have the same sparsity pattern as the slack matrices of -gons. Moreover, it turns out to be a tight bound for the rectangle covering number, a.k.a. the boolean rank, for some (comparing it with the upper bound from [1]).

Second, we slightly improve the upper bound of Fiorini, Rothvoss and Tiwary [9]. Although our approach is equivalent to that of Fiorini et al., both being recursive, our proof is rather different, being purely algebraic as opposed to their geometric approach. Moreover, we are able to reduce the upper bound by one when for some : this is possible by stopping the recursion earlier at a better base case (note that it would be possible to modify the proof of Fiorini et al. to achieve the same bound). We show that for all ,

(1)

Although the improvement is relatively minor, our numerical experiments strongly suggest that this bound is tight; see the discussion at the end of Section 4. Moreover, our bound allows us to close the gap for several -gons as it matches the best known lower bound, for our bound implies that and, for , that ; see Figure 2. (Note that, for , the RRCB was, to the best of our knowledge, never computed prior to this work hence it is also the first time is claimed for .)

The paper is organized as follows. In Section 2, we briefly describe the construction of the slack matrices of regular -gons. In Section 3, we describe our new improved lower bound for the rectangle covering of these matrices, and, in Section 4, we describe our construction that proves the aforementioned upper bound. Then we discuss some directions for further research and conclude in Section 5.

2 The Slack Matrices of Regular -gons

Let us construct the slack matrices of regular -gons. Without loss of generality (w.l.o.g.), we use regular -gons centered at the origin with their vertices located on the unit circle of radius equal to one; see Figure 3 for an illustration with the pentagon.

Figure 3: Illustration for the construction of the slack matrices of regular -gons. In this paper, we assume w.l.o.g. that .

The length of the facets of the regular -gon is given by . The slack between a facet and the th vertex (the th and -th being on the considered facet, and counting along the circle in any direction) is equal to:

(2)

By symmetry, (i) our slack matrices of regular -gons are circulant matrices for which the vector is translated one element to the right on each row, and (ii) the vector satisfies for all . For example, for , we have

(3)

Note that, to the best of our knowledge, the best known lower (resp. upper) bound for is 7 (resp. 8). In this paper, we will improve the upper bound to 7 hence proving that ; see Figure 2.

3 Lower bound for the boolean rank of

In this section, we improve the lower bound on the boolean rank (or, equivalently, the rectangle covering number) for regular -gons. On the way, we derive several new interesting results that could be used to derive other bounds.

Let be an exact nonnegative factorization of of size . In this section, we will use the following notation. Let us define the following subsets of , representing the supports of the rows of and columns of :

Since , and , we have

(4)

If for some , (4) implies that the sparsity pattern of the th row of is contained in the sparsity pattern of the th row of (and similarly for the columns). Therefore, if contains rows whose sparsity patterns are not contained in one another, there are subsets from that form a Sperner family of size , also know as an antichain of size , which is a family of sets that are not contained in one another [22]. By symmetry, the same holds for the columns.

3.1 Sperner theorem and rectangle covering

Sperner theorems bounds the size of an antichain over elements. Let us recall this result and a proof that will be useful later.

Theorem 1.

Let be a set of subsets of . Let also be an antichain, that is, no subset in is contained in another subset in . Then,

(5)

and the bound is tight (take all subsets of size ).

Proof.

([17]) This proof is based on a counting argument using the fact that there are permutations of . Given with elements, there are permutations of whose first elements are in . Because the ’s are not contained in one another, the permutations generated for two different subsets and cannot coincide (otherwise this would imply that or ). Let us also denote the number of sets with elements contained in , that is, , hence . We have

Therefore,

since for all . This completes the proof. ∎

The above result was used to prove that the rectangle covering of the -by- Euclidean distance matrices (with zeros only the diagonal) is the minimum such that ; see [2] and the references therein. This result can actually be generalized for any nonnegative matrix.

Corollary 1 ([4]).

Let be a matrix having rows or columns whose sparsity patterns are not contained in one another. Then,

Proof.

Let have rows with different sparsity patterns. As explained in the introduction of this section, this implies that there are subsets of corresponding to the sparsity patterns of rows of that are not contained in one another. Theorem 1 allows to conclude. ∎

In particular, this result can be applied to the slack matrix of any polytope. In fact, the slack of two different vertices cannot be contained in one another, otherwise it would mean that a vertex is the intersection of a subset of the facets intersecting at another vertex. The same holds for two different facets by polar duality or a similar argument.

Corollary 2.

Let be the slack matrix of a polytope with facets and vertices. Then,

Note that, the results from Corollaries 1 and 2 were already known prior to this work; see, e.g., [14, Cor. 4.13] for a more general result.

In the next section, we apply the same ideas to improve the lower bound for the rectangle covering number of the slack matrices of -gons.

3.2 Improvement for -gons

Let be the slack matrix of a -gons such that if and only if or for ; see Section 2. To simplify the heavy notation , we will assume throughout this section that when represents an index. As before, let be a nonnegative factorization of size of , let denote the support of the th row of () and the support of the th column of (). We have if and only if or , and

Let us try to characterize the size of the sets and that satisfy the above property, where denotes the complement of .

First, we can assume without loss of generality that . In fact, is the largest possible set that does not intersect while having the most intersections with all other sets in (which is the best possible situation since for ).

For the same reason as in Corollary 1, since the rows and columns of have different sparsity patterns, we have that

  • is an antichain.

  • is an antichain, since taking the complement of all the sets in an antichain gives another antichain of the same size.

  • Every set contains at least one element not in the sets for , since for .

Theorem 2.

Let and satisfy (C1-C3) and . Then

Proof.

Let us denote the number of elements in , the number of additional elements in compared to (that is, ) and the number of additional elements in compared to (that is, ). Following the same argument as in Theorem 1, we have that the number of permutations with the elements of in the first positions is given by , of by , and of by . However, between and , there are common permutations (and similarly between and ). Note that these are the only possible repetitions because of (C3). Note also that hence the number of permutations corresponding to are also equal to . Counting all permutations corresponding to and for and accounting for the repetitions, we get

Let us lower bound the left hand side of the above inequality. To do so, we minimize over each term of the sum independently. Noting that and have exactly the same role, we can assume without loss of generality that at a minimum. Removing the index for simplicity, we therefore have to evaluate

In Appendix A, we show that and is an optimal solution. Therefore, dividing the inequality above by and using our lower bound for each term (replacing the ’s with and the ’s with 1), we obtain

from which we get, after simplifications, . ∎

Corollary 3.

Let be the rectangle covering number of the slack matrix of any -gon for , then

Note that the term goes to 1/2 when grows, and we cannot hope to obtain a better bound using our counting argument. In fact, this is the case when there would be no repetitions between the permutations generated from the sets in and ; see the proof of Theorem 2.

The bound from the corollary above also applies to the so-called boolean rank, which is the same as the rectangle coreving number. Comparing our bound with the upper bounds computed in [1, p.145] for small , our bound is tight for ( means from to , that is, ), which was not the case of the previous bound (5) which is tight only for .

4 Explicit nonnegative factorization of slack matrices of regular -gons

In this section, we construct a nonnegative factorization of in a recursive way. The idea is the following. At the first step, a rank-two modification of is performed so that the pattern of zero entries of the constructed matrix therefore looks like a cross (see below for an example on ). This subdivides the matrix into four blocks with a lot of symmetry that implies that the nonnegative rank of one block equals the nonnegative rank of the full matrix. Then, the same scheme is applied to that subblock until the number of columns of the obtained block is smaller than four, which we factorize with a trivial decomposition ( being the identity matrix of appropriate dimension).

Before we rigorously prove that our construction works for any -gon, let us illustrate the idea on the slack matrix of the regular 9-gon form (3). Observe that the entries of the slack matrix on the main diagonal and the diagonal below it are equal to zero. The first step of our construction will make a rank-two correction of the slack matrix so that the same pattern appears: we remove a matrix from the 4-by-4 lower left block of

and another matrix from the positive 4-by-4 block of at the upper right (rows 2 to 5, last 4 columns)

Clearly, the removed matrices are nonnegative since for all . Moreover, we show in the next lemma that they have rank one.

Lemma 1.

The (infinite) matrix

has rank one for any fixed , and .

Proof.

We have that . Choosing any minor with rows and columns (w.l.o.g.), one can check, using algebra with a few trigonometric identities, that the determinant of

is equal to zero for any , , and any . ∎

After these two nonnegative rank-one factors are removed, we obtain

with a pattern of zeros forming a cross. This matrix is highly symmetric and has a lot of redundancy: the last four columns (resp. rows) are copies of the first four. Therefore, if we had a nonnegative factorization of the 5-by-5 upper left block then we would have a nonnegative factorization of the entire matrix with the same nonnegative rank.

To construct that factorization, we apply our strategy recursively: use a rank-two correction to the upper left block to make a cross of zeros appear:

Now, the upper left block has a trivial nonnegative factorization (since it is a 3-by-3 matrix of rank 3) from which we can derive a nonnegative factorization for the full matrix :

Remark 1.

Once the first two rank-one factors have been removed from , the 5-by-5 block could also directly be trivially factorized, and we would obtain