A Universality Theorem for Nested Polytopes

08/06/2019 ∙ by Michael G. Dobbins, et al. ∙ 0

In a nutshell, we show that polynomials and nested polytopes are topological, algebraic and algorithmically equivalent. Given two polytops A⊆ B and a number k, the Nested Polytope Problem (NPP) asks, if there exists a polytope X on k vertices such that A⊆ X ⊆ B. The polytope A is given by a set of vertices and the polytope B is given by the defining hyperplanes. We show a universality theorem for NPP. Given an instance I of the NPP, we define the solutions set of I as V'(I) = {(x_1,...,x_k)∈R^k· n : A⊆conv(x_1,...,x_k) ⊆ B}. As there are many symmetries, induced by permutations of the vertices, we will consider the normalized solution space V(I). Let F be a finite set of polynomials, with bounded solution space. Then there is an instance I of the NPP, which has a rationally-equivalent normalized solution space V(I). Two sets V and W are rationally equivalent if there exists a homeomorphism f : V → W such that both f and f^-1 are given by rational functions. A function f:V→ W is a homeomorphism, if it is continuous, invertible and its inverse is continuous as well. As a corollary, we show that NPP is ∃R-complete. This implies that unless ∃R = NP, the NPP is not contained in the complexity class NP. Note that those results already follow from a recent paper by Shitov. Our proof is geometric and arguably easier.

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1 Introduction

Definition.

In the Nested Polytope Problem (NPP), we are given two polytopes and a number and we ask, whether there exists a polytope with vertices. To be more precise the inner polytope is specified by its vertices and the outer polytope is specified by its facets. Given an instance , we denote by

the set of solutions. Here denotes the convex hull of the points . Given a permutation , we can for every solution get a new solution denoted by . Note that has a lot of symmetries as every permutation of the vertices of a valid solution yields again a valid solution. We say two solutions are permutation-equivalent if there exists a permutation of the vertices, so that . We denote this by . We define the normalized solution space by

It is not a priori clear that can be interpreted as a subset of . And for some instances , this will not be the case. However, for the instances that we produce it is. For us every instance has a set of disjoint segments associated to it. We will show that on each segment must lie exactly one vertex in any valid solution. Let be an order on . Then we can think of simply as the vertices of the intermediate polytope given in the order and thus .

Rational-Equivalence.

On a very high-level, a universality theorem states that we can represent any objects of type  by an object of type  preserving property . In our case, objects of type  are just bounded algebraic varietes, which we will formally define below. They are very versatile as they can encode many different mathematical objects of interest in a straight-forward fashion. Instances of the NPP are the objects of type . At last, we want to preserve algebraic and topological properties. In this paragraph, we define the notion of rational-equivalence, which preserve both [34].

Let be a finite set of polynomials with . Then we define the variety of as

We say is bounded, if there is a ball such that .

Two varieties and are rationally equivalent if there exists a homeomorphism such that both and are given by rational functions. A function is a homeomorphism, if it is continuous, invertible and its inverse is continuous as well. The function is rational, if it can be component-wise described as the ratio of polynomials. We denote rational-equivalence by . Note that the composition of two homeomorphisms is a homeomorphism. Similarly, the composition of two rational functions is rational. Next to algebraic and topological properties, we preserve also algorithmic properties. To state this properly, we will introduce the complexity class in the next paragraph.

Existential Theory of the Reals.

In the study of geometric problems, the complexity class plays a crucial role, connecting purely geometric problems and Real Algebraic Geometry. Whereas NP is defined in terms of existentially quantified Boolean variables, deals with existentially quantified real variables.

Consider a first-order formula over the reals that contains only existential quantifiers,

where are real-valued variables and is a quantifier-free formula involving equalities and inequalities of integer polynomials. The algorithmic problem Existential Theory of the Reals (ETR) takes such a formula as an input and asks whether it is satisfiable. The complexity class consists of all problems that reduce in polynomial time to ETR. Many problems in combinatorial geometry and geometric graph representation naturally lie in this class, and furthermore, many have been shown to be -complete, e.g., stretchability of a pseudoline arrangement [25, 27, 32], recognition of segment intersection graphs [23] and disk intersection graphs [26], computing the rectilinear crossing number of a graph [5], etc. For surveys on , see [33, 11, 25]. A recent proof that the Art Gallery Problem is -complete [1] provides the framework we follow in our proof. See also [6, 36, 11, 32, 33, 25, 20, 24, 26, 31, 10, 9] for a small selection of -complete problems.

Results.

We show a universality theorem for the NPP. Note that the result is implied by a recent result of Shitov [34] about Non-negative Matrix Factorization and an old reduction due to Cohen and Rotblum [14]. Thus we attribute the result to Shitov.

Theorem 1 (Universality Shitov [34].).

For every bounded variety exists an instances of the Nested Polytope Problem such that .

In this paper, we give a direct proof that does not use either of the two above papers. The main ideas of our proof are simple geometric constructions. This implies that polynomial equations have a solution space that is topologically and algebraically equivalent to solution spaces given by the NPP. To illustrate the strength of the statement, we highlight give one algebraic corollary and one topological example.

Corollary 2 (Algebraic Consequences).

Let be two algebraic field extensions of . Then there exists an instance of the NPP such that there is a solution in , but not in .

This implies for instance the result by [12], who showed that there is an instance of the Nested Polytope Problem that requires irrational coordinates.

Example 3 (Topological Consequences).

Let be a torus, then there is an instance of the NPP such that the solution space is homeomorphic to .

Note that the polynomial equation

describes a torus with the two radii and . To see the last corollary, simply apply Theorem 1 on the variety given by , with .

As all the steps involved to show this universality theorem take polynomial time to execute we can infer the algorithmic complexity of the NPP.

Corollary 4 (Shitov [34]).

The Nested Polytope Problem is -complete.

In the rest of the introduction, we survey the literature on the NPP and the closely related problem of Non-negative Matrix Factorization.

Proof Overview.

The proof consists of two parts. In Section 2, we show that a certain very simple set of polynomial equations is already -complete and admits the desired universality property. In a nutshell, we only allow only the constraints and .

In the second part, in Section 3, we are encoding those constraints in the NPP. The first idea is to enforce certain vertices to lie on specific line segments. Those vertices are encoding variables. It is very easy to build polytopes that encode the two constraints explained above. The main technical challenge is to ”stick” those smaller building blocks together to a ”big one”. This is easy, if you are used to work with polytopes in higher dimensions. In our description, we do not assume the reader to have that familiarity. See also Figure 1

Figure 1: Red indicates the outer polytope, blue the inner polytope, and green the nested polytope. Left: The nested polytope must have one vertex on the left bottom edge. Middle: Assuming that one nested vertex is on the left edge and one at the top edge, then the two vertices encode inversion. Right: Assuming that each green vertex is forced to be on its vertical segment, then the blue vertex enforces the constraint .
Related Work on Nested Polytopes.

To the best of our knowledge the NPP was first mentioned by Silio in 1979 [37], who could find an time algorithm in the case that the outer and inner polytope are convex polygons in the plane with and vertices respectively. Additionally, Silio restricts to the case . The motivation of Silio came from a connection to Stochastic Sequential Machines.

Independently, Victor Klee suggested the same problem as was pointed out in several papers [16, 2, 18, 30, 15], the first of them dating back to 1985. In particular, the NPP appears as one of the open problem in the Computational Geometry Column  [30]. The main motivation of those early papers used to be simplification of a given polytope, see Figure 2.

Figure 2: Using an algorithm for the nested polygon problem it is possible to attain a simplified version of the previous polygon.

Among the first results is an algorithm for the nested convex polygon problem [2]. On the lower bounds side, Das and Joseph showed NP-hardness for the NPP in dimension three [16, 18, 17, 15].

In 1995, Suri and Mitchell were able to reduce the NPP to a set cover problem, by loosing only a factor of . Their motivation to study the NPP came from separating geometric objects. Using the greedy approximation scheme for set cover they attain an -approximation algorithm that runs in time ( number of facets of inner and outer polytope, dimension). This was consequently improved by Brönnimann and Goodrich [7] as the first application in their seminal paper on -nets. The key observation is that the set-cover system described by Suri and Mitchell has bounded VC-dimension. Their algorithm runs in and gives an -approximation ( size of the optimal solution) . Independently Clarkson [13]

found a similar approximation algorithm using techniques from linear programming.

Interestingly, the NPP has close relations to Non-negative Matrix Factorization. We define NMF, explain the history and this relation in the next paragraph.

Non-Negative Matrix Factorization.

In a parallel line of research the NMF is explored, with the earliest mentioning, we found, in 1973 [4]. The non-negative matrix factorization is defined as follows. Given a matrix and a number , we say that , for matrices and , is a non-negative matrix factorization of inner dimension . We denote by the set of non-negative real numbers. We denote by the non-negative rank, which is the smallest inner dimension, for which a non-negative matrix factorization exists.

While it is said that NMF

has many applications in image processing, machine learning, dimension reduction and clustering, in theory it is most famous for the relationship to extension complexity. Given a polytope

, its extension complexity, is the smallest number such that there is a polytope on facets such that there is a linear projection from to . Yannakakis showed in its seminal paper [41] (Roughly saying that on needs an exponential size symmetric LP to solve the travelling salesperson problem.) that the extension complexity of a polytope is the non-negative rank of its slack matrix. See also [21] for the lower bound for non-symmetric LPs.

For us most relevant is a reduction from the NMF to NPP by Cohen and Rothblum [14]. To the delight of the reader, we repeat this reduction in the appendix.

Lemma A (Cohen Rothblum [14]).

Let be a matrix and be a number. Let be the convex hull intersected with and the positive orthant intersected with . Then as an instance to the NPP is equivalent to as NMF.

In 2009, Vavasis [40] showed that exact-non-negative matrix factorization is equivalent to the intermediate simplex problem. In particular, their results imply that the NPP is already hard, if the intermediate polytope is restricted to be a simplex. In a similar way, Gillis and Glineur [22] in 2010 showed that restricted non-negative matrix factorization is equivalent to NPP.

Although, it is easy to encode NMF as an algebraic decision problem the huge number of variables makes it algorithmically infeasible. In 2012, Arora Ge, Kannan and Moitra [3] found an algorithm that runs in polynomial time for every fixed . They also showed that there is no , assuming ETH. In 2016 Moitra [29] improved the upper bound and gave an algorithm for NMF. Note that those results translate immediately to results about NPP, due to the reductions mentioned above.

Chistikov, Kiefer, Marušić, Shirmohammadi, Worrell have shown that the optimal solution of the Nested Polytope Problem requires irrational coordinates already in dimension and . It is an open problem, if the intermediate-simplex problem requires irrational coordinates as well. This was also shown in parallel by Shitov [35]. Furthermore, Shitov showed a universality result for NMF, very similar to our result. In particular, his result implies -completeness of both NMF and the NPP.

2 Encoding ETR

In this section, we define the algorithmic problem and the complexity class both called the Exitential Theory of the Reals. For distinction, the algorithmic problem is denoted by ETR and the complexity class by .

An instance of ETR is a well-formed logical formula of the form

The subformula is quantifier free. It has polynomial equations and (strict) inequalities as atomic formulas. Those atomic formulas can be combined in any boolean way. For example:

Strictly speaking, we are only allowed to use variables and the symbols

However, we interpret as and as and so on. We are asking if there is an assignment of real numbers to the variables such that the formula becomes true. Note that the definition in the introduction and the more precise definition here are equivalent, although this may not be obvious.

It is not a priory clear that there even exists an algorithm which can decide this problem. Due to Tarski’s Quantifier Elimination [38], we know that this question can be decided. Even more, we know that the problem can be solved in polynomial space, due to Canny [8]. The complexity class is defined as the set of algorithmic problems that can be reduced in polynomial time to ETR.


An ETR-INV-system of size

is a vector of

real variables together with a system of linear and quadratic equations of the form

The solution space of an ETR-INV-system is the set of all vectors in which satisfy the equations of , where we do allow the possibility of an empty solution space. (Note that the solution space of an ETR-INV-system is a real semi-algebraic set.) It was shown in [1, Lemma 12] that the problem of determining whether an ETR-INV-system has a non-empty solution space is -complete. (The original formulation of ETR-INV-system included the equation , but this equation can be obtained by and ) Although, this was not pointed out directly, if we follow the reduction it is easy to observe that all steps are rationally-equivalent.

Lemma 5 (Universality Inversion).

Let be a finite set of polynomials , with bounded solution space. There is an instance of ETR-INV-system such that

Proof Sketch..

In almost every step of the reduction a new variable and a new constraint is introduce. All other variables are left as they are. For example, , where is an old variable and is a new variable. Of course, there is the assumption that . We see that , which determines that the new and the old system of equations are rationally equivalent. To be explicit, the following mapping

is a homeomorhpism, it is rational and its inverse is rational as well.

Note that there are two exceptions. The inequality is replaced by . Note that this step does not preserve homotopy as the two sets

and

do not have the same number of connected components. However, we restrict ourselves to systems of polynomial equations.

The second exception is when all variables are scaled down to a small range. Note that the sets

and

are not rationally equivalent. But again, this does not apply to us, as we assume that the initial solution space is bounded. ∎

While ETR-INV-system is the right intermediate problem to show that the Art Gallery Problem is -complete, for the purpose of this paper however, it will be more convenient to work with a slight modification of ETR-INV-system, which we introduce now.

An ETR-INV-array of size is an -by- matrix of variables together with a system of linear and quadratic equations of the form

(Note that the linear equations relate variables in the same row and the quadratic equations relate variables in the same column.)

The solution space of an ETR-INV-array is defined similarly as for ETR-INV-system and is a semi-algebraic subset of . We now have the following lemma.

Lemma 6 (Uninversality of ETR-INV-array).

Let be an ETR-INV-system on variables. There exists an ETR-INV-array of size such that the solution spaces of and are rationally-equivalent. The description complexity of is linear in .

Proof.

Let us denote by the variables of . First note that we can assume without loss of generality that every variable in is in at most one inversion constraint involved. Otherwise and , implies and we can replace everywhere by and forget about . Note that this preserves rational equivalence.

We denote the variables of by , for and we write them into the array as follows

We want that all constraints in for the variables hold in for the corresponding variables. The remaining variables in are supposed to be completely determined. We introduce the linear constraint

for every . Let us first consider linear constraints of the form

in . We introduce the linear constraint,

Note that this implies

This encodes all linear constraints. Now let us consider the quadratic constraints, involving two different variables. Note that we denote the pairs of constraints as

For every , we are adding the constraints

Similarly, we add the constraints

This enforces and . Furthermore, we add the constraints

This enforces , as desired.

Let us now consider the special case of . Recall that this is equivalent to . We introduce the constraints

This is equivalent to .

Note that all the ’s variables and some of the variables are still completely unconstrained. Add any linear constraint of the form with an already used variable to them, so that they are uniquely determined, by one of the . We have that the set of constraints on the ’s and one the ’s are exactly the same.

Let use denote by the solution space of and by the solution space of . We have to show that and are rationally-equivalent. To this end we define the mapping

Let in . Then we define . We define . Each and is either or , if the index was contained in a pair in . All other variables are of the form or , for some . Note that is bijective, as the set of constraints onto the ’s and ’s are the same. The mapping is continuous and rational by definition. The inverse mapping is simply given by

This is also a continuous and rational mapping. Note that the number of constraints and variables of is linear in the number of variables and constraints of . This finishes the proof. ∎

3 Building the polytopes

The main goal of this section is to prove the following lemma.

Lemma 7.

Let be an ETR-INV-array of size . There exists convex polytopes such that there exists a nested polytope with vertices such that the solution spaces are rationally-equivalent.

Remark 8.

The polytopes in Lemma 7 are actually contained in a hyperplane in , and are -dimensional. The outer polytope will have vertices and is defined by hyperplanes. The vertex description and facet description of the outer polytope will be given in Subsection 3.3.1.

The inner polytope has vertices in common with the outer polytope together with an additional vertices that lie on certain 2-faces of the outer polytope. Finally, for each equation in the ETR-INV-array we add one additional vertex to the inner polytope which will lie on certain faces of the outer polytope. The vertex description of the inner polytope is given in Subsection 3.3.2.

3.1 Two geometric observations

Here we state two simple geometric observations that are used for the “gadgets” needed in our construction of the polytopes of Lemma 7.

3.1.1 The linear equations

Let be a set of affinely independent points in . For let and define the prism as

For define the point as

Finally, for define points as

where . A simple calculation (left to the reader) gives us the following.

Observation 9.

if and only if .

3.1.2 The quadratic equation

In the plane , let be a point on the line and let be a point on the line , where . A simple calculation (left to the reader) gives us the following.

Observation 10.

The origin if and only if .

3.2 A basic outline of the construction

We now give an outline of the construction of the polytopes in Lemma 7, without giving explicit coordinates, but rather focusing on the three “gadgets” that will be used to encode the three types of equations in . (We will give precise coordinates in Subsection 3.3.)

3.2.1 The outer polytope

To build the outer polytope we start with an “orthogonal frame” spanning , consisting of mutually orthogonal segments of equal length all meeting in a common endpoint. Note that the convex hull of these segments form an -dimensional simplex. (When we eventually add coordinates, the length of these segments will be 3 units, each one parametrizing the closed interval .) We now take distinct copies of the orthogonal frame, , , , each one translated into “independent dimensions” so that their union now lives in (Note that the affine span of the union will be -dimensional.) We label the segments of these orthogonal frames as

such that the segments are all parallel.

We now take the outer polytope to be the convex hull of . It is straight-forward to show that is an -dimensional polytope with vertices. In what follows, for each and , the “second half” of segment , parametrizing the interval , will correspond to the variable in the ETR-INV-array . The segments will play an auxiliary role which we describe next.

3.2.2 Building the inner polytope: Enforcing vertices to segments

The first step in building the inner polytope is to enforce the following.

Property 11.

Let be a nested polytope, with vertices and . For every and , the segment contains exactly one vertex of , which we denote by .

(More specifically, each segment of the orthogonal frame will contain exactly one vertex from in its “second half”, thus encoding a value in the interval .) This can be done as follows. Fix indices and , and consider segment and its parallel copy , which are edges of a -dimensional face of the outer polytope . Define the point to be the unique point in this 2-face such that segment is mapped to the second half of its parallel copy by central projection through . Similarly, we define the analogous point in the -face of spanned by the segment and its parallel copy . (See Figure 3.)

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Figure 3: The vertices of any nested polytope (marked in red) must include the endpoints of segments and , while the last vertex, , must be contained in the segment restricted to the interval .

At this stage of the construction the inner polytope will consist of the orthogonal frames and together with the points for all and . Moreover, if is a nested polytope, with vertices and , then must contain the orthogonal frames and . This accounts for of the vertices, as all (the same for the .) have one end point in common. Futermore, must contain one vertex in each of the segments of the orthogonal frames . This accounts for the remaining vertices. Thus Property 11 is satisfied, and we let denote the unique vertex of which is contained in the (second half of the) segment , which we associate with the variable .

3.2.3 Building the inner polytope: Encoding

In order to enforce the relation , we add a new vertex to the inner polytope as follows. We consider the rectangular 2-face of the outer polytope spannced by the segements and . Define to be the point in this 2-face such that is contained in the convex hull of the vertices and of the nested polytope (satisfying Property 11) if and only if the associated variables . (The unique point exists by Observation 9 by letting be the endpoints of and be the endpoints of . See Figure 4.)

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Figure 4: The vertices and contain the point in their convex hull if and only if the associated variables satisfy the equation

Let us now add the vertex to , and consider a nested polytope satisfying Property 11. It is easily shown that the only possible way that can contain the point , is if this point is contained in . In other words, if satisfies Property 11, then the associated variables satisfy the equation .

3.2.4 Building the inner polytope: Encoding

Enforcing the relation is similar to the previous case, and we add a new vertex to the inner polytope as follows. We consider the triangluar prism spanned by the segments , , and , which is a 3-face of the outer polytope .

Define to be the point in this 3-face such that is contained in the convex hull of the vertices , , and of the nested polytope (satisfying Property 11) if and only if the associated variables . (The unique point exists by Observation 9 by letting be the endpoints of , be the endpoints of , and be the endpoints of . See Figure 5.)

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Figure 5: The vertices , , and contain the point if and only if the associated variables satisfy the equation .

Let us now add the vertex to , and consider a nested polytope satisfying Property 11. It is easily seen that the only possible way that can contain the point is if this point is contained in . In other words, if satisfies Property 11, then the associated variables satisfy the equation .

3.2.5 Building the inner polytope: Encoding

In order to enforce the relation we add a new vertex to the inner polytope as follows. Consider the triangular 2-face of spanned by segments and . Note that the two segments belong to the same orthogonal frame and thus share an endpoint and are orthogonal to one another by definition. We can coordinatize the plane containing this 2-face such that the segment is parametrized by and the segment is parametrized by . We then define to be the origin with respect to this coordinate system. It follows from Observation 10 that the vertices and contain the point in their convex hull if and only if the associated coordinates satisfy the equation . (See Figure 6.)

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Figure 6: The vertices and contain the point if and only if the associated variables satisfy the equation .

Let us now add the point to , and consider a nested polytope satisfying Property 11. As before, it is easily shown that the only way that can contain the point is if this point is contained in . In other words, if satisfies Property 11, then the associated variables satisfy the equation .

3.3 Explicit coordinates

We now give the explicit coordinates to the construction in the previous section. Let denote the standard basis in and set

3.3.1 The outer polytope

We start by giving the vertices of the outer polytope . First define

and for , let

This defines the orthogonal frame by setting the segment .

Similarly, define

and for , let

This defines the orthogonal frame by setting the segment .

Next we define the orthogonal frames . For , let

and for , , let

For every this defines the orthogonal frame by setting .

Finally, for , , and set and . The outer polytope is now defined as . Equivalently, is the set of in the affine hyperplane

that satisfy the following linear constraints,

Observe that is a -dimensional convex polytope with vertex set and facets defined by the above constraints. Furthermore, is a subset of the vertices of the axis-aligned box , and therefore is in convex position and form the vertices of . To see that is -dimensional, we simply note that

is an affinely independent set of size .

3.3.2 The inner polytope

We now define vertices of the inner polytope . These will consist of the points , defined above, together with some additional points.

For and define the points and that ensure Property 11 are defined as

and

The point which encodes the equation is defined as