Exponential Lower Bounds for Polytopes in Combinatorial Optimization

Introduction

Since the advent of the simplex method , linear programming has become a prominent tool for solving optimization problems in practice. On the theoretical side, LPs can be solved in polynomial time via either the ellipsoid method or interior point methods .

In 1986–1987 there were attempts to prove P $=$ NP by giving a polynomial-size LP that would solve the traveling salesman problem (TSP). Due to the large size and complicated structure of the proposed LP for the TSP, it was difficult to show directly that the LP was erroneous. In a groundbreaking effort to refute all such attempts, Yannakakis proved that every symmetric LP for the TSP has exponential size (see for the journal version). Here, an LP is called symmetric if every permutation of the cities can be extended to a permutation of all the variables of the LP that preserves the constraints of the LP. Because the proposed LP for the TSP was symmetric, it could not possibly be correct.

In his paper, Yannakakis left as a main open problem the question of proving that the TSP admits no polynomial-size LP, symmetric or not. We solve this question by proving a super-polynomial lower bound on the number of inequalities in every LP for the TSP. We also prove such unconditional super-polynomial lower bounds for the maximum cut and maximum stable set problems. Therefore, it is impossible to prove P = NP by means of a polynomial-size LP that expresses any of these problems. Our approach is inspired by a close connection between semidefinite programming reformulations of LPs and one-way quantum communication protocols that we introduce here.

The idea of representing the set of feasible solutions of a problem by a polytope forms the basis of a standard and powerful methodology in combinatorial optimization, see, e.g., .

Extended Formulations and Extensions

Resuming the discussion above (and assuming that the problem is a minimization problem), we have $\min\{f(x)\mid x\in X\}=\min\{f(x)\mid x\in P\}=\min\{f(x)\mid Ax\leqslant b\}$, where $Ax\leqslant b$ is any linear description of $P$. This turns any given instance of the combinatorial optimization problem into an LP, however over an implicit system of constraints the LP is potentially large since it has at least one inequality per facet of $P$. In fact, even for polynomially solvable problems, the associated polytope $P$ may have an exponential number of facets.

Here, we often restrict to EFs in slack form, that is, containing only equalities and one nonnegativity inequality per additional variable:

The proof of the factorization theorem (Theorem 3) shows that this can be done without loss of generality, see Remark 1. In the following we put EFs in slack form to ease the generalization to arbitrary cones. Notice that the size of an EF in slack form can equivalently be defined as the number of additional variables since the only inequalities are from $y\geqslant\mathbf{0}$.

The Impact of Extended Formulations

EFs have pervaded the areas of discrete optimization and approximation algorithms for a long time. For instance, Balas’s disjunctive programming , the Sherali-Adams hierarchy , the Lovász-Schrijver closures , lift-and-project , and configuration LPs are all based on the idea of working in an extended space. Recent surveys on EFs in the context of combinatorial optimization and integer programming are .

Symmetry Matters

/1-Polytopes with Large Extension Complexity

The Factorization Theorem

Yannakakis discovered that the extension complexity of a polytope $P$ is determined by certain factorizations of an associated matrix, called the slack matrix of $P$, that records for each pair $(F,v)$ of a facet $F$ and vertex $v$, the algebraic distance of $v$ to a valid hyperplane supporting $F$. Defining the nonnegative rank of a matrix $M$ as the smallest natural number $r$ such that $M$ can be expressed as $M=TU$ where $T$ and $U$ are nonnegative matrices (i.e., matrices whose elements are all nonnegative) with $r$ columns (in case of $T$) and $r$ rows (in case of $U$), respectively, it turns out that the extension complexity of every polytope $P$ is exactly the nonnegative rank of its slack matrix.

We point out that this result generalizes to any slack matrix of the polytope, which may contain additional rows corresponding to faces $F$ of $P$ which are not facets and/or additional columns corresponding to points $v$ of $P$ that are not vertices. This fact is used in the proof of our lower bounds on extension complexity, starting with Theorem 7.

This factorization theorem led Yannakakis to explore connections between EFs and communication complexity. Let $S$ denote the slack matrix of the polytope $P$. He proved that: (i) every deterministic communication protocol of complexity $k$ computing $S$ gives rise to an EF of $P$ of size at most $2^{k}$; (ii) the nondeterministic communication complexity of the support matrix of $S$ (i.e., the binary matrix that has 0-entries exactly where $S$ is 0) yields a lower bound on (the base-2 logarithmAll logarithms in this paper are in base $2$. of) the extension complexity of $P$, or more generally, the nondeterministic communication complexity of the support matrix of every nonnegative matrix $M$ yields a lower bound on (the base-2 logarithm of) the nonnegative rank of $M$.The classical nondeterministic communication complexity of a binary communication matrix is defined as $\lceil\log B\rceil$, where $B$ is the minimum number of monochromatic 1-rectangles that cover the matrix, see . This last quantity is also known as the rectangle covering bound. It is easy to see that the rectangle covering bound of the support matrix of any matrix $M$ lower bounds the nonnegative rank of $M$ (see Theorem 4 below).

Tighter Communication Complexity Connection

Faenza et al. proved that the base-$2$ logarithm of the nonnegative rank of a matrix equals, up to a small additive constant, the minimum complexity of a randomized communication protocol with nonnegative outputs that computes the matrix in expectation. In particular, every EF of size $r$ can be regarded as such a protocol of complexity $\log r+O(1)$ bits that computes a slack matrix in expectation.

The Clique vs. Stable Set Problem

A notoriously hard open question is to determine the communication complexity (in the deterministic or nondeterministic sense) of the clique vs. stable set problem. (For recent results that explain why this question is hard, see .) The best lower bound to this day is due to : they obtained a $\frac{6}{5}\log n-O(1)$ lower bound. Furthermore, they state a graph-theoretical conjecture that, if true, would imply a $\Omega(\log^{2}n)$ lower bound, and hence settle the communication complexity of the clique vs. stable set problem. Moreover it would give a worst-case $n^{\Omega(\log n)}$ lower bound on the extension complexity of stable set polytopes. However, a solution to the Huang-Sudakov conjecture seems far away.

Factorization Theorem for General Cones

2 Our Contribution

Our contribution in this paper is two-fold.

In addition to simultaneously settling the above-mentioned open problems of Yannakakis and Rothvoß, our results provide a lower bound on the extension complexity of stable set polytopes that goes much beyond what is implied by the Huang-Sudakov conjecture (thanks to the fact that we consider a different part of the slack matrix). Although our lower bounds are strong, unconditional and apply to explicit polytopes that are well-known in combinatorial optimization, they have very accessible proofs.

Second, we generalize the tight connection between linearIn this paragraph, and later in Section 4, an EF (in the sense of the previous section) is called a linear EF. The use of adjectives such as “linear”, “semidefinite” or “conic” will help distinguishing the different types of EFs. EFs and classical communication complexity found by Faenza et al. to a tight connection between semidefinite EFs and quantum communication complexity.After a first version of this paper appeared, Jain et al. [38, Theorem 2] have used this notion of PSD rank to characterize the number of qubits of communication between Alice and Bob needed to generate a shared probability distribution. We show that any rank-$r$ PSD factorization of a (nonnegative) matrix $M$ gives rise to a one-way quantum protocol computing $M$ in expectation that uses $\log r+O(1)$ qubits and, conversely, that any one-way quantum protocol computing $M$ in expectation that uses $q$ qubits results in a PSD factorization of $M$ of rank $2^{q}$. Via the semidefinite factorization theorem, this yields a characterization of the semidefinite extension complexity of a polytope in terms of the minimum complexity of (one-way) quantum protocols that compute the corresponding slack matrix in expectation.

Then, we give a complexity $\log r+O(1)$ quantum protocol for computing a nonnegative matrix $M$ in expectation, whenever there exists a rank-$r$ matrix $N$ such that $M$ is the entry-wise square of $N$. This implies in particular that every $d$-dimensional polytope with 0/1 slacks has a semidefinite EF of size $O(d)$.

Finally, we obtain an exponential separation between classical and quantum protocols that compute our specific matrix $M=M(n)$ in expectation. On the one hand, our quantum protocol gives a rank-$O(n)$ PSD factorization of $M$. On the other hand, the nonnegative rank of $M$ is $2^{\Omega(n)}$ because the nondeterministic communication complexity of the support matrix of $M$ is $\Omega(n)$. Thus we also obtain an exponential separation between PSD rank and nonnegative rank.

We would like to point out that the lower bounds on the extension complexity of polytopes established in Section 3 were obtained by first finding an efficient PSD factorization or, equivalently, an efficient one-way quantum communication protocol for the matrix $M=M(n)$. In this sense our classical lower bounds stem from quantum considerations somewhat similar in style to . See for a survey of this line of work.

We would also like to point out that the fact that a matrix $M$ with a rank-$r$ entrywise square-root has a PSD-rank at most $r+O(1)$, which follows from Theorem 16, was also obtained by Gouveia, Parrilo and Thomas , independently (since their results were not publicly available at the time we performed our research) and in a different context. Also, after a preprint of our paper had appeared, we learned that Klauck et al. lee:communication had independently found a matrix (similar but not quite the same as ours) with an exponential separation between PSD rank and nonnegative rank.

3 Other Related and Subsequent Work

Yannakakis’s paper has deeply influenced the TCS community. In addition to the works cited above, it has inspired a whole series of papers on the quality of restricted approximate EFs, such as those defined by the Sherali-Adams hierarchies and Lovász-Schrijver closures starting with ( for the journal version), see, e.g., .

After the conference version of our paper appeared, there has been a lot of follow-up work, including on approximations. Braun et al. developed a general framework for studying the power of approximate EFs, independent of specific hierarchies. In particular, via lower bounds on the extension complexity of approximations of the cut polytope, they showed that linear programs for approximating Max-Clique to within a factor $n^{1/2-\epsilon}$ need size at least $2^{\Omega(n^{\epsilon})}$. Similarly, they show the existence of a spectrahedron of small size that cannot be approximated by any LP with a polynomial number of inequalities within a factor of $n^{1/2-\epsilon}$. Braverman and Moitra used methods from information complexity to show the same size lower bound even for approximation factor $n^{1-\epsilon}$; Braun and Pokutta subsequently simplified and generalized their result and Braun et al. show that the amortized log nonnegative rank is characterized by information. Such inapproximability results should be contrasted with Håstad’s famous result that it is hard to approximate Max-Clique to within a factor $n^{1-\epsilon}$: Håstad’s result gives is a lower bound for all algorithms approximating Max-Clique and is conditional on the unproven assumption that RP $\neq$ NP, while the results of and are geometric statements about the nonexistence of polynomial-size extended formulations.

Braun, Fiorini and Pokutta analyze the average-case polyhedral complexity of the maximum stable set problem showing that the extension complexity of the stable set polytope is high for almost all graphs. Pokutta and Van Vyve proved lower bounds on extension complexity for the knapsack problem, and Avis and Tiwary proved lower bounds for the subset-sum and three-dimensional matching problems, as well as others. Kaibel and Weltge gave a more direct proof of the lower bound for the cut polytope, via bounding the size of the largest rectangle in the slack matrix, however, they still use the same set of $2^{n}$ valid constraints that we use here (Lemma 6).

Chan et al. prove super-polynomial lower bounds on approximate EFs for MAX CSPs (constraint satisfaction problems). In particular, they prove that every $(2-\varepsilon)$-approximate (linear) EF for Max-Cut has $n^{\Omega\left(\frac{\log n}{\log\log n}\right)}$ size. This is striking because the celebrated approximation algorithm of Goemans and Williamson is based on a $\Theta(n)$-size semidefinite EF with an approximation factor of at most $1.14$. Again, the result of Chan et al. on Max-Cut matches the algorithmic hardness of the problem Khot et al. , which assumes the Unique Games Conjecture.

Rhothvoß proves that the matching polytope has extension complexity $2^{\Omega(n)}$, solving the second part of the main open problem in . This is the first time such a strong bound is obtained for a polytope over which one can optimize in polynomial time. Rothvoß’s groundbreaking result implies in particular that the extension complexity of the TSP polytope is $2^{\Omega(n)}$, thus going beyond our $2^{\Omega(\sqrt{n})}$ lower bound.

Not much is known yet about lower bounds on semidefinite EFs. Extending the work of Rothvoß, Briët, Dadush and Pokutta show that most 0/1 polytopes (i.e., polytopes that are the convex hull of a random subset of $\{0,1\}^{d}$) need exponentially large semidefinite EFs. Fawzi and Parrilo give exponential lower bounds on the size of semidefinite EFs of explicit polytopes in a restricted setting, where the underlying cone is not the full PSD cone but rather a product of fixed-dimensional PSD cones. Lee and Theis obtain polynomial lower bounds based on the support pattern of slack matrices.

Finally, Fiorini et al. use the notion of conic extensions and its relation to communication complexity to study generalized probabilistic theories, which are different from the usual classical or quantum-mechanical theories, and show that all polynomially-definable 0/1-polytopes have small extension complexity with respect to the completely positive cone.

4 Organization

The discovery of our lower bounds on extension complexity crucially relied on finding the right matrix $M$ and the right polytope whose slack matrix contains $M$. In our case, we found these through a connection with quantum communication. However, these quantum aspects are not strictly necessary for the resulting lower bound proof itself. Hence, in order to make the main results more accessible to those without background or interest in quantum computing, we start by giving a purely classical presentation of those lower bounds.

In Section 2 we define our matrix $M$ and lower bound the nondeterministic communication complexity of its support matrix. In Section 3 we embed $M$ in the slack matrix of the cut polytope in order to lower bound its extension complexity; further reductions then give lower bounds on the extension complexities of the stable set, and TSP polytopes. In Section 4 we establish the equivalence of PSD factorizations of a (nonnegative) matrix $M$ and one-way quantum protocols that compute $M$ in expectation, and give an efficient quantum protocol in the case where some entry-wise square root of $M$ has small rank. This is then used to provide an exponential separation between quantum and classical protocols for computing a matrix in expectation (equivalently, an exponential separation between nonnegative rank and PSD rank). Concluding remarks are given in Section 5.

A Simple Matrix with Large Rectangle Covering Bound

In this section we consider the following $2^{n}\times 2^{n}$ matrix $M=M(n)$ with rows and columns indexed by $n$-bit strings $a$ and $b$, and real nonnegative entries:

Note for later reference that $M_{ab}$ can also be written as

For a given matrix, a rectangle is the Cartesian product of a set of row indices and a set of column indices. In it was shown that an exponential number of (monochromatic) rectangles are needed to cover all the 1-entries of the support matrix of $M$. Equivalently, the corresponding function $f:\{0,1\}^{n}\times\{0,1\}^{n}\rightarrow\{0,1\}$ has nondeterministic communication complexity of $\Omega(n)$ bits. For the sake of completeness we repeat the proof here:

We use the following result from [47, Example 3.22 and Section 4.6], which is essentially due to Razborov :

There exist sets $A,B\subseteq\{0,1\}^{n}\times\{0,1\}^{n}$ and probability distribution $\mu$ on $\{0,1\}^{n}\times\{0,1\}^{n}$ such that all $(a,b)\in A$ have $a^{\intercal}b=0$, all $(a,b)\in B$ have $a^{\intercal}b=1$, $\mu(A)=3/4$, and there are constants $\alpha,\delta>0$ (independent of $n$) such that for all rectangles $R$,

(For sufficiently large $n$, $\alpha=1/135$ and $\delta=0.017$ will do.)

Since the $R_{i}$ are 1-rectangles, they cannot contain elements from $B$. Hence $\mu(R_{i}\cap B)=0$ and $\mu(R_{i}\cap A)\leqslant 2^{-\delta n}/\alpha$. However, since all elements of $A$ are covered by the $R_{i}$, we have

Strong Lower Bounds on Extension Complexity

Here we use the matrix $M=M(n)$ defined in the previous section to prove that the (linear) extension complexity of the cut polytope of the $n$-vertex complete graph is $2^{\Omega(n)}$, i.e., every (linear) EF of this polytope has an exponential number of inequalities. Then, via reductions, we prove super-polynomial lower bounds for the stable set polytopes and the TSP polytopes. To start, let us define more precisely the slack matrix of a polytope. For a matrix $A$, let $A_{i}$ denote the $i$th row of $A$ and let $A^{j}$ denote the $j$th column of $A$.

an extended formulation (EF) of $P$ is a linear system in variables $(x,y)$ such that $x\in P$ if and only if there exists $y$ satisfying the system;

the extension complexity of $P$ is the minimum size (i.e., number of inequalities) of an EF of $P$.

We are ready to state Yannakakis’s factorization theorem.

$P$ has an extension of size at most $r$ (that is, with at most $r$ facets);

$P$ has an EF of size at most $r$ (that is, with at most $r$ inequalities).

It should be clear that (ii) implies (iii). We prove that (i) implies (ii), and then that (iii) implies (i).

First, consider a rank-$r^{*}$ nonnegative factorization $S=TU$ of the slack matrix of $P$, where $r^{*}\leqslant r$. Notice that we may assume that no column of $T$ is zero, because otherwise $r^{*}$ can be decreased. We claim that $P$ is the image of

under the projection $\pi_{x}:(x,y)\mapsto x$ onto the $x$-space. We see immediately that $\pi_{x}(Q)\subseteq P$ since $Ty\geqslant\mathbf{0}$. To prove the inclusion $P\subseteq\pi_{x}(Q)$, it suffices to remark that for each point $v_{j}\in V$ the point $(v_{j},U^{j})$ is in $Q$ since

Since no column of $T$ is zero, $Q$ is a polytope. Moreover, $Q$ has at most $r^{*}\leqslant r$ facets, and is thus an extension of $P$ of size at most $r$. This proves that (i) implies (ii).

We would like to emphasize that we will not restrict the slack matrix to have rows corresponding only to the facet-defining inequalities. This is not an issue since appending rows corresponding to redundantAn inequality of a linear system is called redundant if removing the inequality from the system does not change the set of solutions. inequalities does not change the nonnegative rank of the slack matrix. This fact was already used in the second part of the previous proof.

Theorem 3 shows in particular that we can lower bound the extension complexity of $P$ by lower bounding the nonnegative rank of its slack matrix $S$; in fact it suffices to lower bound the nonnegative rank of any submatrix of the slack matrix $S$ corresponding to an implied system of inequalities. To that end, Yannakakis made the following connection with nondeterministic communication complexity. Again, we include the (easy) proof for completeness.

If $M=TU$ is a rank-$r$ nonnegative factorization of $M$, then $S$ can be written as the sum of $r$ nonnegative rank-$1$ matrices:

2 Cut and Correlation Polytopes

For all $a\in\{0,1\}^{n}$, the inequality

There exists some constant $C>0$ such that, for all $n$,

In their follow-up work, Kaibel and Weltge proved that one can take $C=\log(3/2)\approx 0.58$.

3 Stable Set Polytopes

Recall that a polytope $Q$ is an extension of a polytope $P$ if $P$ is the image of $Q$ under a linear projection.

Consider the complete graph $K_{n}$ with vertex set $V_{n}:=[n]$. For each vertex $i$ of $K_{n}$ we create two vertices labeled $ii,\overline{ii}$ in $H_{n}$ and an edge between them. Let us label the edges of $K_{n}$ in the following way. The edge between vertices $i$ and $j$ with $i<j$ gets the label $ij$. Now, for each edge $ij$ of $K_{n},$ we add to $H_{n}$ four vertices labeled $ij,\overline{ij},\underline{ij},\overline{\underline{ij}}$ and all possible six edges between them. We further add the following eight edges to $H_{n}$:

See Fig. 1 for an illustration. The number of vertices in $H_{n}$ is $2n+4{n\choose 2}.$

Our next lemma establishes simple monotonicity properties of the extension complexity used in our reduction.

Let $P$, $Q$ and $F$ be polytopes. Then the following hold:

The first part is obvious because every extension of $F$ is in particular an extension of $P$. For the second part, notice that a slack matrix of $F$ can be obtained from the (facet-vs-vertex) slack matrix of $Q$ by deleting columns corresponding to vertices not in $F$. Now apply Theorem 3. ∎

Using previous results, we can prove the following result about the worst-case extension complexity of the stable set polytope.

4 TSP Polytopes

To prove the lemma we start with constructing a graph $G_{n}$ with $q=O(n^{2})$ vertices such that the tours of $G_{n}$ correspond to the $n\times n$ rank-$1$ binary symmetric matrices $bb^{\intercal}$, where $b\in\{0,1\}^{n}$. This is done in three steps:

define a 3SAT formula $\phi_{n}$ with $n^{2}$ variables such that the satisfying assignments of $\phi_{n}$ bijectively correspond to the matrices $bb^{\intercal}$, where $b\in\{0,1\}^{n}$;

construct a directed graph $D_{n}$ with $O(n^{2})$ vertices such that each directed tour of $D_{n}$ defines a satisfying assignment of $\phi_{n}$, and conversely each satisfying assignment of $\phi_{n}$ has at least one corresponding directed tour in $D_{n}$;

modify the directed graph $D_{n}$ into an undirected graph $G_{n}$ in such a way that the tours of $G_{n}$ bijectively correspond to the directed tours of $D_{n}$.

Step (i). For defining $\phi_{n}$ we use Boolean variables $C_{ij}\in\{0,1\}$ for $i,j\in[n]$ and let

The four clauses $(C_{ii}\lor C_{jj}\lor\overline{C_{ij}})$, $(C_{ii}\lor\overline{C_{jj}}\lor\overline{C_{ij}})$, $(\overline{C_{ii}}\lor C_{jj}\lor\overline{C_{ij}})$ and $(\overline{C_{ii}}\lor\overline{C_{jj}}\lor C_{ij})$ model the equation $C_{ij}=C_{ii}\land C_{jj}$. Hence, $C\in\{0,1\}^{n\times n}$ satisfies $\phi_{n}$ if and only if there exists $b\in\{0,1\}^{n}$ such that $C_{ij}=b_{i}\land b_{j}$ for all $i,j\in[n]$, or in matrix language, $C=bb^{\intercal}$.

Step (ii). To construct a directed graph $D_{n}$ whose directed tours correspond to the satisfying assignments of $\phi_{n}$ we use the standard reduction from 3SAT to HAMPATH .

Next we connect the gadgets corresponding to the variables by identifying $t_{k}$ with $s_{k+1}$ for $1\leqslant k<n^{2}$. Finally, we add a directed edge from $t_{n^{2}}$ to $s_{1}$. Figure 4 illustrates the final directed graph obtained.

To see why the directed tours of the final directed graph $D_{n}$ define satisfying assignments of our Boolean formula $\phi_{n}$, observe that each directed tour of $D_{n}$ encodes a truth assignment to the $n^{2}$ variables depending on which way the corresponding chains are traversed. Because a directed tour visits every node and because the node $w_{m}$ corresponding to a clause can be visited only if we satisfy it, the truth assignment satisfies $\phi_{n}$. Conversely, every satisfying assignment of $\phi_{n}$ yields at least one directed tour in $D_{n}$. (If the $m$th clause is satisfied by the value of more than one variable, we visit $w_{m}$ only once, from the chain of the first variable whose value makes the clause satisfied).

The final theorem in this section follows from Theorem 7, Lemmas 9 and 11, using an argument similar to that used in the proof of Theorem 10.

Quantum Communication and PSD Factorizations

In this section we explain the connection with quantum communication. This yields results that are interesting in their own right, and also clarifies where the matrix $M$ of Section 2 came from.

2 Quantum Protocols

A one-way quantum protocol with $r$-dimensional messages can be described as follows. On input $i$, Alice sends Bob an $r$-dimensional state $\rho_{i}$. On input $j$, Bob measures the state he receives with a POVM $\{E^{j}_{\theta}\}$ for some nonnegative values $\theta$, and outputs the result. We say that such a protocol computes a matrix $M$ in expectation, if the expected value of the output on respective inputs $i$ and $j$, equals the matrix entry $M_{ij}$. Analogous to the equivalence between classical protocols and nonnegative factorizations of $M$ established by Faenza et al. , such quantum protocols are essentially equivalent to PSD factorizations of $S$:

A one-way quantum protocol with $r$-dimensional messages that computes $M$ in expectation, gives a rank-$r$ PSD factorization of $M$.

A rank-$r$ PSD factorization of $M$ gives a one-way quantum protocol with $(r+1)$-dimensional messages that computes $M$ in expectation.

so the protocol indeed computes $M$ in expectation. ∎

We obtain the following corollary which summarizes the characterization of semidefinite EFs:

For a polytope $P$ with slack matrix $S$, the following are equivalent:

the slack matrix $S$ has a rank-$r$ PSD factorization;

there exists a one-way quantum communication protocol with $(r+1)$-dimensional messages (i.e., using $\lceil\log(r+1)\rceil$ qubits) that computes $S$ in expectation (for the converse we consider $r$-dimensional messages).

3 A General Upper Bound on Quantum Communication

Now we provide a quantum protocol that efficiently computes a nonnegative matrix $M$ in expectation, whenever there is a low rank matrix $N$ whose entry-wise square is $M$.

Let $M$ be a matrix with nonnegative real entries, $N$ be a rank-$r$ matrix of the same dimensions such that $M_{ij}=N^{2}_{ij}$. Then there exists a one-way quantum protocol using $(r+1)$-dimensional pure-state messages that computes $M$ in expectation.

By Corollary 15, it suffices to give a rank-$r$ PSD factorization of $M$. To this end, let $t_{i},u_{j}$ be $r$-dimensional real vectors such that $N_{ij}=t_{i}^{\intercal}u_{j}$; such vectors exist because $N$ has rank $r$. Define $r\times r$ PSD matrices $T_{i}:=t_{i}t_{i}^{\intercal}$ and $U^{j}:=u_{j}u_{j}^{\intercal}$. Then

hence we have a rank-$r$ PSD factorization of $M$. ∎

4 Quantum vs Classical Communication, and PSD vs Nonnegative Factorizations

We now give an example of an exponential separation between quantum and classical communication in expectation, based on the matrix $M$ of Section 2. This result actually preceded and inspired the results in Section 3.

For the classical lower bound, note that a protocol that computes $M$ in expectation has positive probability of giving a nonzero output on input $a,b$ if and only if $M_{ab}>0$. With a message $m$ in this protocol we can associate a rectangle $R_{m}=A\times B$ where $A$ consists of all inputs $a$ for which Alice has positive probability of sending $m$, and $B$ consists of all inputs $b$ for which Bob, when he receives message $m$, has positive probability of giving a nonzero output. Together these rectangles will cover exactly the nonzero entries of $M$. Accordingly, a $c$-bit protocol that computes $M$ in expectation induces a rectangle cover for the support matrix of $M$ of size $2^{c}$. Theorem 1 lower bounds the size of such a cover by $2^{\Omega(n)}$, hence $c=\Omega(n)$. ∎

Together with Theorem 14 and the equivalence of randomized communication complexity (in expectation) and nonnegative rank established in , we immediately obtain an exponential separation between nonnegative rank and PSD rank.

Concluding Remarks

In addition to proving the first unconditional super-polynomial lower bounds on the size of linear EFs for the cut polytope, stable set polytope and TSP polytope, we demonstrate that the rectangle covering bound can prove strong results in the context of EFs. In particular, it can be super-polynomial in the dimension and the logarithm of the number of vertices of the polytope, settling an open problem of .

An important problem also left open in is whether the perfect matching polytope has a polynomial-size linear EF. Yannakakis proved that every symmetric EF of this polytope has exponential size, a striking result given the fact that the perfect matching problem is solvable in polynomial time. He conjectured that asymmetry also does not help in the case of the perfect matching polytope. Because it is based on the rectangle covering bound, our argument does not yield a super-polynomial lower bound on the extension complexity of the perfect matching polytope. This question was recently answered in the affirmativeby Rothvoß posted on arXiv a proof of the fact that the extension complexity of the perfect matching polytope is $2^{\Omega(n)}$. This groundbreaking result is based on a general lower bound called the hyperplane separation bound, which was used implicitly, e.g., in .

As mentioned at the end of the introduction, the new connections developed have already inspired much follow-up research in particular about approximate EFs. Here are two concrete questions left open for future work: (i) find a slack matrix that has an exponential gap between nonnegative rank and PSD rank; (ii) prove that the cut polytope has no polynomial-size semidefinite EF (that would rule out SDP-based algorithms for optimizing over the cut polytope, in the same way that this paper ruled out LP-based algorithms).

We thank Kota Ishihara for carefully reading the manuscript and pointing out an error in a previous version of the text. We thank Monique Laurent for information about hypermetric inequalities, and the three anonymous STOC’12 referees as well as one JACM referee for suggesting improvements to the text. Sebastian Pokutta would like to thank Alexander Martin for the inspiring discussions and support. Ronald de Wolf thanks Giannicola Scarpa and Troy Lee for useful discussions.

Samuel Fiorini acknowledges support from the Actions de Recherche Concertées (ARC) fund of the French community of Belgium. Serge Massar acknowledges support from the European Commission under the projects QCS (Grant No. 255961) and QALGO (Grant No. 600700). Hans Raj Tiwary was postdoctoral researcher of the Fonds National de la Recherche Scientifique (F.R.S.–FNRS). Ronald de Wolf was partially supported by a Vidi grant from the Netherlands Organization for Scientific Research (NWO), by ERC Consolidator grant QPROGRESS, and by the European Commission under the projects QCS (Grant No. 255961) and QALGO (Grant No. 600700).

References

Appendix A Background on Polytopes

If $P$ is not full-dimensional, these statements have to be adapted as follows. Every (finite) system describing $P$ contains all the facet-defining inequalities of $P$, up to scaling by positive numbers and adding an inequality that is satisfied with equality by all points of $P$. Conversely, a linear description of $P$ can be obtained by picking one inequality per facet and adding a system of equalities describing $\text{aff}(P)$.

For more background on polytopes and polyhedra, see the standard reference .