The Lipton-Markakis-Mehta algorithm (LMM) is a well known method for computing approximate Nash equilibria in normal form games . The key idea behind their technique is to prove that there exist approximate Nash equilibria where both players use simple strategies.
Suppose that we have a convex set
defined by vectorsthrough . A vector is -uniform if it can be written as a sum of the form where each is a non-negative integer and .
Since there are at most -uniform vectors, one can enumerate all -uniform vectors in time. For approximate equilibria in bimatrix games, Lipton, Markakis, and Mehta showed that for every there exists an -Nash equilibrium where both players use -uniform strategies where , and so they obtained a quasi-polynomial approximation scheme (QPTAS) for finding an -Nash equilibrium.
Their proof of this fact uses a sampling argument. Every bimatrix game has an exact Nash equilibrium (NE), and each player’s strategy in this NE is a probability distribution. If we sample from each of these distributionstimes, and then construct new -uniform strategies using these samples, then when there is a positive probability the new strategies form an -NE. So by the probabilistic method, there must exist a -uniform -NE.
The sampling technique has been widely applied. It was initially used by Althöfer  in zero-sum games, before being applied to non-zero sum games by Lipton, Markakis, and Mehta . Subsequently, it was used to produce algorithms for finding approximate equilibria in normal form games with many players , sparse bimatrix games , tree polymatrix , and Lipschitz games . It has also been used to find constrained approximate equilibria in polymatrix games with bounded treewidth .
At their core, each of these results uses the sampling technique in the same way as the LMM algorithm: first take an exact solution to the problem, then sample from this solution times, and finally prove that with positive probability the sampled vector is an approximate solution to the problem. The details of the proofs, and the value of , are often tailored to the specific application, but the underlying technique is the same.
The existential theory of the reals.
In this paper we ask the following question: is there a broader class of problems to which the sampling technique can be applied? We answer this by providing a sampling theorem for the existential theory of the reals. The existential theory of the reals consists of existentially quantified formulae using the connectives over polynomials compared with the operators . For example, each of the following is a formula in the existential theory of the reals.
Given a formula in the existential theory of the reals, we must decide whether the formula is true, that is, whether there do indeed exist values for the variables that satisfy the formula.
The complexity class ETR is defined to be all problems that can be reduced in polynomial time to the existential theory of the reals. It is known that ETR PSPACE , and NP ETR since the problem can easily encode Boolean satisfiability. However, the class is not known to be equal to either PSPACE or NP, and it seems to be a distinct class of problems between the two. Many problems are now known to be ETR-complete, including various problems involving constrained equilibria in normal form games with at least three players [6, 7, 8, 9, 22].
In this paper we propose the approximate existential theory of the reals (-ETR), where we seek a solution that approximately satisfies the constraints of the formula. We show a subsampling theorem for a large fragment of -ETR, which can be used to obtain PTASs and QPTASs for the problems that lie within it. We believe that this will be useful for future research: instead of laboriously reproving subsampling results for specific games, it now suffices to simply write a formula in -ETR and then apply our theorem to immediately get the desired result. To exemplify this, we prove several new QPTAS and PTAS results using our theorem.
Our first result is actually that -ETR = ETR, meaning that the problem of finding an approximate solution to an ETR formula is as hard as finding an exact solution. However, this result crucially relies on the fact that ETR formulas can have solutions that are doubly-exponentially large. This motivates the study of constrained -ETR, where the solutions are required to lie within a given convex set.
Our main theorem (Theorem 3.1) gives a subsampling result for constrained -ETR. It states that if the formula has an exact solution, then it also has a -uniform approximate solution, where the value of depends on various parameters of the formula, such as the number of constraints and the number of variables. The theorem allows for the formula to be written using tensor constraints, which are a type of constraint that is useful in formulating game-theoretic problems.
The consequence of the main theorem is that, when various parameters of the formula are constant (see Corollary 1), we are able to obtain a QPTAS for approximating the existential theory of the reals. Specifically, this algorithm either finds an approximate solution of the constraints, or verifies that no exact solution exists. In many game theoretic applications an exact solution always exists, and so this algorithm will always find an approximate solution.
It should be noted that we are not just applying the well-known subsampling techniques in order to derive our main theorem. Our main theorem incorporates a new method for dealing with polynomials of degree , which prior subsampling techniques were not able to deal with.
Our theorem can be applied to a wide variety of problems. In the game theoretic setting, we prove new results for constrained approximate equilibria in normal form games, and approximating the value vector of a Shapley game. We also show optimization results. Specifically, we give approximation algorithms for optimizing polynomial functions over a convex set, subject to polynomial constraints. We also give algorithms for approximating eigenvalues and eigenvectors of tensors. Finally, we apply the theorem to some problems from computational geometry.
2 The Existential Theory of the Reals
Let , , , be distinct variables, which we will treat as a vector . A term of a multivariate polynomial is a function , where are non negative integers and . A multivariate polynomial is a function , where each is a term as defined above, and is a constant.
We now define Boolean formulae over multivariate polynomials. The atoms of the formula are polynomials compared with , and the formula itself can use the connectives .
The existential theory of the reals consists of every true sentence of the form , where is a Boolean formula over multivariate polynomials of through .
Given a Boolean formula , the ETR problem is to decide whether is a true sentence in the existential theory of the reals. We will say that has constraints if it uses operators from the set in its definition.
The approximate ETR.
In the approximate existential theory of the reals, we replace the operators with their approximate counterparts. We define the operators and with the interpretation that holds if and only if and if and only if . The operators and are defined analogously.
We do not allow equality tests in the approximate ETR. Instead, we require that every constraint of the form should be translated to before being weakened to .
We also do not allow negation in Boolean formulas. Instead, we require that all negations are first pushed to atoms, using De Morgan’s laws, and then further pushed into the atoms by changing the inequalities. So the formula would first be translated to before then being weakened to .
The approximate existential theory of the reals consists of every true sentence of the form , where is a negation-free Boolean formula using the operators over multivariate polynomials of through .
Given a Boolean formula , the -ETR problem asks us to decide whether is a true sentence in the approximate existential theory of the reals, where the operators are used.
Our first result is that if no constraints are placed on the value of the variables, that is if each can be arbitrarily large, then -ETR = ETR for all values of . We show this via a two way reduction between -ETR and ETR. The reduction from -ETR to ETR is trivial, since we can just rewrite each constraint as , and likewise for the other operators.
For the other direction, we show that the ETR-complete problem , which asks us to decide whether a system of multivariate polynomials has a shared root, can be formulated in -ETR. Here we rely on a result of Schaefer and Stefankovic , which showed that has a solution if and only if there is a point such that for all , where is the number of bits used to represent the polynomials. To formulate the problem in -ETR, we blow-up the instance by multiplying each polynomial by a doubly-exponentially large number that is bigger than . The number can be constructed by a polynomially-sized formula that uses repeated squaring. So if we write down the constraint in -ETR, then this implies that and therefore . Thus, via the lemma of Schaefer and Stefankovic, we can formulate in the -ETR. The full details of this reduction are given in Appendix 0.A.
-ETR = ETR for all .
In our negative result for unconstrained -ETR, we abused the fact that variables could be arbitrarily large to construct the doubly-exponentially large number . So, it makes sense to ask whether -ETR gets easier if we constrain the problem so that variables cannot be arbitrarily large.
In this paper, we consider -ETR problems that are constrained by a convex set in . For vectors we use to denote the set containing every vector that lies in the convex hull of through . In the constrained -ETR, we require that the solution of the -ETR problem should also lie in the convex hull of through .
Given a Boolean formula and vectors , the constrained -ETR problem asks us to decide whether
Note that, unlike the constraints used in , the convex hull constraints are not weakened. So the resulting solution , , , must actually lie in the convex set.
3 Approximating Constrained -Etr
To state our main theorem, we will use a certain class of polynomials where the coefficients are given as a tensor. This will be particularly useful when we apply our theorem to certain problems, such as normal form games. To be clear though, this is not a further restriction on the constrained -ETR problem, since all polynomials can be written down in this form.
The variables of the polynomials we will study will be -dimensional vectors denoted as , where will denote the -th element of vector . The coefficients of the polynomials will be a tensor denoted by . Given a tensor , we denote by its element with coordinates on the tensor’s dimensions , respectively, and by we denote the maximum absolute value of these elements. We define the following two classes of polynomials.
Simple tensor multivariate.
We will use denote an STM polynomial with variables where each variable , is applied times on tensor that defines the coefficients. Tensor has dimensions with indices each. We will say that an STM polynomial is of maximum degree , if . Here is an example of a degree 2 simple tensor polynomial with two variables:
This polynomial itself is written as follows.
Tensor multivariate. A tensor multivariate (TMV) polynomial is the sum over a number of simple tensor multivariate polynomials. We will use to denote a tensor multivariate polynomial with vector variables, which is formally defined as
where the exponents depend on , and is the number of simple multivariate polynomials. We will say that has length if it is the sum of STM polynomials, and that it is of degree if .
-ETR with tensor constraints.
We focus on -ETR instances where all constraints are of the form , where is an operator from the set . Recall that each TMV constraint considers vector variables. We consider the number of variables used in (denoted as ) to be the number of vector variables used in the TMV constraints. So the value of used in our main theorem may be constant if only a constant number of vectors are used, even if the underlying -ETR instance actually has a non-constant number of variables. For example, if and and are -dimensional probability distributions and and are tensors, the TMV constraint has three variables, degree 1, length two; though the underlying problem has variables.
Note that every -ETR constraint can be written as a TMV constraint, because all multivariate polynomials can be written down as a TMV polynomial. Every term of a TMV can be written as a STM polynomial where the tensor entry is non zero for exactly the combination of variables used in the term, and otherwise. Then a TMV polynomial can be constructed by summing over the STM polynomial for each individual term.
The main theorem.
Given an -ETR formula , we define to be a Boolean formula in which every approximate constraint is replaced with its exact variant, meaning that every instance of is replaced with , and likewise for the other operators.
Our main theorem is as follows.
Let be an -ETR instance with vector variables and multivariate-polynomial constraints each one of maximum length and maximum degree , constrained by a convex set defined by . Let be the maximum absolute value of the coefficients of constraints of , and let . If has a solution in , then has a -uniform solution in where
Consequences of the main theorem.
Our main theorem gives a QPTAS for approximating a fragment of -ETR. The total number of -uniform vectors in a convex set is . So, if the parameters , , , , and are all constant, then our main theorem tells us that the total number of -uniform vectors is , where is the number of constraints. So if we enumerate each -uniform vector , we can check whether holds, and if it does, we can output . If no -uniform vector exists that satisfies , then we can determine that has no solution. This gives us the following result.
Let be an -ETR instance constrained by the convex set defined by . If , , , , and are constant, and is polynomial, then we have an algorithm that runs in time that either finds a solution to , or determines that has no solution.
If is constant and is polynomial then this gives a PTAS, while if and are polynomial, then this gives a QPTAS.
In Section 5 we will show that the problem of approximating the best social welfare achievable by an approximate Nash equilibrium in a two-player normal form game can be written down as a constrained -ETR formula where , , , and are constant. It has been shown that, assuming the exponential time hypothesis, this problem cannot be solved faster than quasi-polynomial time [11, 19], so this also implies that constrained -ETR where , , , and are constant cannot be solved faster than quasi-polynomial time unless the exponential time hypothesis is false.
Many -ETR problems are naturally constrained by sets that are defined by the convex hull of exponentially many vectors. The cube is a natural example of one such set. Brute force enumeration does not give an efficient algorithm for these problems, since we need to enumerate vectors, and is already exponential. However, our main theorem is able to provide non-deterministic polynomial time algorithms for these problems.
This is because each -uniform vector is, by definition, the convex combination of at most of the vectors in the convex set, and this holds even if is exponential. So, provided that is polynomial, we can guess the subset of vectors that are used, and then verify that the formula holds. This is particularly useful for problems where
always has a solution, which is often the case in game theory applications, since it places the approximation problem in NP, whereas computing the exact solution may be ETR-complete.
Let be an -ETR instance constrained by the convex set defined by . If , , , , , are polynomial, then there is a non-deterministic polynomial time algorithm that either finds a solution to , or determines that has no solution. Moreover, if is guaranteed to have a solution, then the problem of finding an approximate solution for is in NP.
A theorem for non-tensor formulas.
One downside of Theorem 3.1 is that it requires that the formula is written down using tensor constraints. We have argued that every ETR formula can be written down in this way, but the translation introduces a new vector-variable for each variable in the ETR formula. When we apply Theorem 3.1 to obtain PTASs or QPTASs we require that the number of vector variables is at most polylogarithmic, and so this limits the application of the theorem to ETR formulas that have at most polylogarithmically many variables.
The following theorem is a sampling result for -ETR with non-tensor constraints, which is proved in Appendix 0.B.
Let be an instance constrained over the convex set defined by . Let be the number of constraints used in , Let , let be the largest constant coefficient used in , let be the number of terms used in , and let be the maximum degree of the polynomials in . If has a solution in , then has a -uniform solution in where
The key feature here is that the number of variables does not appear in the formula for , which allows the theorem to be applied to some formulas for which Theorem 3.1 cannot. However, since the theorem does not allow tensor constraints, its applicability is more limited because the number of terms will be much larger in non-tensor formulas. For example, as we will see in Section 5, we can formulate bimatrix games using tensor constraints over constantly many vector variables, and this gives a result using Theorem 3.1. No such result can be obtained via Theorem 3.2, because when we formulate problem without tensor constraints, the number of terms used in the inequalities becomes polynomial.
4 The Proof of the Main Theorem
In this section we prove Theorem 3.1. Before we proceed with the technical results let us provide a roadmap. We begin by considering two special cases, which when combined will be the backbone of the proof of the main theorem.
Firstly, we will show how to deal with problems where every constraint of the Boolean formula is a multilinear polynomial, which we will define formally later. We deal with this kind of problems using Hoeffding’s inequality and the union bound, which is similar to how such constraints have been handled in prior work.
Then, we study problems where the Boolean formula consists of a single degree polynomial constraint. We reduce this kind of problems to a constrained -ETR problem with multilinear constraints, so we can use our previous result to handle the reduced problem. Degree polynomials have not been considered in previous work, and so this reduction is a novel extension of sampling based techniques to a broader class of -ETR formulas.
Finally, we deal with the main theorem: we reduce the original ETR problem with multivariate constraints to a set of problems with a single standard degree constraint, and then we use the last result to derive a bound on .
Problems with multilinear constraints.
We begin by considering constrained -ETR problems where the Boolean formula consists of tensor-multilinear polynomial constraints. We will use to denote a tensor-multilinear polynomial with variables and coefficients defined by tensor of size . Formally,
We will use to denote the maximum entry of tensor in the absolute value sense and to denote the infinite norm of the convex set that constrains the variables.
The following lemma is proved in Appendix 0.C. The proof uses Hoeffding’s inequality and the union bound, and is similar to previous applications of the sampling technique.
Let be a Boolean formula with variables and tensor-multilinear polynomial constraints and let be a convex set in the variables space. If the constrained ETR problem defined by and has a solution, then the constrained -ETR problem defined by and has a uniform solution where
Problems with a standard degree constraints.
We now consider constrained -ETR problems with exactly one tensor polynomial constraint of standard degree . We will use to denote a standard degree tensor-polynomial with coefficients defined by the tensor . Here, identical vectors are applied on A. Formally,
The following lemma is proved in Appendix 0.D. To prove the lemma we consider the variable to be defined as the average of variables. This allows us to “break” the standard degree tensor polynomial to a sum of multilinear tensor polynomials and to a sum of not-too-many multivariate polynomials. Then, the choice of allows us to upper bound the error occurred by the multivariate polynomials by . Then, we observe that in order to prove the lemma we can write the sum of multilinear tensor polynomials as an -ETR problem with variables and roughly multilinear constraints. This allows us to use Lemma 1 to complete the proof.
Let be a Boolean formula with variable and one tensor-polynomial constraint of standard degree , and let be a convex set. If the constrained ETR problem defined by and has a solution, then the constrained -ETR problem defined by and has a -uniform solution where
Problems with simple multivariate constraints.
We now assume that we are given a constraint--ETR problem defined by a Boolean formula of tensor simple multilinear polynomial constraints and a convex set . As before and let be the maximum absolute value of the coefficients of the constraints. We will say that the constraints are of maximum degree if is the maximum degree among all variables. The following lemma is proved in Appendix 0.E. The idea is to rewrite the problem as an equivalent problem with standard degree constraints and then apply Lemmas 2 and 1 to derive the bound for .
Let be a Boolean formula with variables and simple tensor-multivariate polynomial constraints of maximum degree and let be a convex set in the variables space. If the constrained ETR problem defined by and has a solution, then the constrained -ETR problem defined by and has a uniform solution where
The proof of Theorem 3.1.
Assume that is a solution for . Consider now a multivariate constraint of defined by . Firstly, we replace this constraint by
Then, replace Constraint (1) by constraints of the form
where are the simple tensor multivariate polynomials consists of. By the triangle inequality we get that if all constraints given by (2) hold, then Constraint (1) holds as well. Hence, we can reduce the problem to an equivalent problem with the same variables and constraints that all of them are simple tensor multivariate polynomials. So, we can apply Lemma 3 where we replace with and with . This completes the proof of the theorem. ∎
We now show how our theorems can be applied to derive new approximation algorithms for a variety of problems.
Constrained approximate Nash equilibria.
A constrained Nash equilibrium is a Nash equilibrium that satisfies some extra constraints, like specific bounds on the payoffs of the players. Constrained Nash equilibria attracted the attention of many authors, who proved NP-completeness for two-player games [23, 14, 6] and ETR-completeness for three-player games [6, 7, 8, 9, 22] for constrained exact Nash equilibria.
Constrained approximate equilibria have been studied, but so far only lower bounds have been derived [2, 25, 11, 19, 18]. It has been observed that sampling methods can give QPTASs for finding constrained approximate Nash equilibria for certain constraints in two player games .
By applying Theorem 3.1, we get the following result for games with a constant number of players: Any property of an approximate equilibrium that can be formulated in -ETR where , , , and are constant has a QPTAS. This generalises past results to a much broader class of constraints, and provides results for games with more than two players, which had not previously been studied in this setting. The details of this result are given in Appendix 0.F.
Shapley’s stochastic games  describe a two-player infinite-duration zero-sum game. The game consists of states. Each state specifies a two-player matrix game where the players compete over: (1) a reward (which may be negative) that is paid by player two to player one, and (2) a probability distribution over the next state of the game. So each round consists of the players playing a bimatrix game at some state , which generates a reward, and the next state of the game. The reward in round is discounted by , where is a discount factor. The overall payoff to player 1 is the discounted sum of the infinite sequence of rewards generated during the course of the game.
Shapley showed that these games are determined, meaning that there exists a value vector , where is the value of the game starting at state . A polynomial-time algorithm has been devised for computing the value vector of a Shapley game when the number of states is constant . However, since the values may be irrational, this algorithm needs to deal with algebraic numbers, and the degree of the polynomial is , so if is even mildly super-constant, then the algorithm is not polynomial.
Shapley showed that the value vector is the unique solution of a system of polynomial optimality equations, which can be formulated in ETR. Any approximate solution of these equations gives an approximation of the value vector, and applying Theorem 3.1 gives us a QPTAS. This algorithm works when , which is a value of that prior work cannot handle. The downside of our algorithm is that, since we require the solution to be bounded by a convex set, the algorithm only works when the value vector is reasonably small. Specifically, the algorithm takes a constant bound , and either finds the approximate value of the game, or verifies that the value is strictly greater than . The details of the algorithm are given in Appendix 0.G.
Our framework can provide approximation schemes for optimization problems with one vector variable with polynomial constraints over bounded convex sets. Formally,
where are polynomials with respect to vector ; for example , where is an matrix, subject to and . We will call the polynomials solution-constraints. Optimization problems of this kind received a lot of attention over the years [15, 16, 17, 21].
For optimization problems, we sample from the solution that achieves the maximum when we apply Theorem 3.1, in order to prove that there is a -uniform solution that is close to the maximum. Our algorithm enumerates all -uniform profiles, and outputs the one that maximizes the objective function. Using this technique, Theorem 3.1 gives the following results.
There is a PTAS if is a STM polynomial of maximum degree independent of , the number of solution-constraints is independent of , and .
There is QPTAS if is an STM polynomial of maximum degree up to , the number of solution-constraints is , and .
To the best of our knowledge, the second result is new. The first result was already known, however it was proven using completely different techniques: in  it was proven for the special case of degree two, in  it was extended to any fixed degree, and alternative proofs of the fixed degree case were also given in [16, 17]. We highlight that in all of the aforementioned results solution constraints were not allowed. Note that unless NP=ZPP there is no FPTAS for quadratic programming even when the variables are constrained in the simplex . Hence, our results can be seen as a partial answer to the important question posed in : “What is a complete classification of functions that allow a PTAS?”
Our framework provides quasi-polynomial time algorithms for deciding the existence of approximate eigenvalues and approximate eigenvectors of tensors in , where the elements are bounded by a constant, where the solutions are required to be in a convex set. In  it is proven that there is no PTAS for these problems when the domain is unrestricted. To the best of our knowledge this is the first positive result for the problem even in this, restricted, setting. The details of the algorithm are given in Appendix 0.H.
Finally, we note that our theorem can be applied to problems in computational geometry, although the results are not as general as one may hope. Many problems in this field are known to be ETR-complete, including, for example, the Steinitz problem for 4-polytopes, inscribed polytopes and Delaunay triangulations, polyhedral complexes, segment intersection graphs, disk intersection graphs, dot product graphs, linkages, unit distance graphs, point visibility graphs, rectilinear crossing number, and simultaneous graph embeddings. We refer the reader to the survey of Cardinal  for further details.
All of these problems can be formulated in -ETR, and indeed our theorem does give results for these problems. However, our requirement that the bounding convex set be given explicitly limits their applicability. Most computational geometry problems are naturally constrained by a cube, so while Corollary 2 does give NP algorithms, we do not get QPTASs unless we further restrict the convex set. In Appendix 0.I we formulate QPTASs for the segment intersection graph and the unit disk intersection graph problems when the solutions are restricted to lie in a simplex. While it is not clear that either problem has natural applications that are restricted in this way, we do think that future work may be able to derive sampling theorems that are more tailored towards the computational geometry setting.
P. Spirakis wishes to dedicate this paper to the memory of his late father in law Mathematician and Professor Dimitrios Chrysofakis, who was among the first in Greece to work on tensor analysis.
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Appendix 0.A Proof of Theorem 2.1
In this section we will show that unconstrained -ETR = ETR for all . Every -ETR instance can be trivially reduced in polynomial time to an ETR instance by replacing each constraint of the form with the constraint , and likewise translating and to their exact counterparts.
It is less obvious that every ETR formula can be reformulated as an -ETR formula. We will prove this by modifying a technique of Schaefer and Stefankovic . They considered the following problem, which asks us to find a shared root of a system of polynomials.
Definition 4 ()
Given a system of multi-variate polynomials , decide whether there exists an such that for all .
Schaefer and Stefankovic showed that this problem is ETR-complete.
Theorem 0.A.1 ()
We will reduce to -ETR. Let be an instance of , and let be the number of bits needed to represent this instance. We define . The following lemma was shown by Schaefer and Stefankovic.
Lemma 4 ()
Let be an instance of . If there does not exist an such that for all , then for every there exists an such that .
In other words, if the instance of is not solvable, then one of the polynomials will always be bounded away from by at least .
The first task is to build an -ETR formula that ensures that a variable satisfies . This can be done by the standard trick of repeated squaring, but we must ensure that the -inequalities do not interfere with the process. We define the following formula over the variables , where all of the following constraints are required to hold.
|for all .|
In other words, this requires that , and . So we have , and hence . Note that the size of this formula is polynomial in the size of .
Given an instance of we create the following -ETR instance , where all of the following are required to hold.
|for all ,||(3)|
|for all ,||(4)|
where the final inequality is implemented using the construction given above.
is satisfiable if and only if has a solution.
First, let us assume that has a solution. This means that there exists an such that for all . Note that clearly satisfies Inequalities 3 and 4, while Inequality 5 can be satisfied by fixing to be any number greater than . So we have proved that is satisfiable.
On the other hand, now we will assume that satisfies . Note that we must have
and hence for all . But Lemma 4 states that this is only possible in the case where has a solution. ∎
This completes the proof of Theorem 2.1.
Appendix 0.B Proof of Theorem 3.2
We will use the following theorem of Barman.
Theorem 0.B.1 ()
Let with . For every and every there exists a -uniform vector such that .
The following lemma shows that if we take two vectors and that are close in the norm, then for all polynomials the value of cannot be too large.
Let be a multivariate polynomial over with degree and let for some constant . For every pair of vectors with we have:
Consider a term of , which can without loss of generality be written as , where it could be the case that any number of ’s are the same. We have
where the second to last four lines use the fact that ’s, and are all less than or equal to .
Next consider a term of of degree . This can be written similarly to the aforementioned term. Then . Since there are many terms in , we therefore have that
We now apply this to prove Theorem 3.2.
Proof (of Theorem 3.2)
Appendix 0.C Proof of Lemma 1
Let be a solution for . Since we assume the is a convex set of any can be written as a convex combination of the ’s, i.e., , where for every , and . Observe, corresponds to a probability distribution over , where vector is drawn with probability , and can be thought of as the mean of . So, we can “sample” a point by sampling over s according to the probability that define this point.
For every , let be a -uniform vector sampled independently from . To prove the lemma, we will show that, because of the choice of , with positive probability the sampled vectors satisfy every constraint of the -ETR problem. Then, by the probabilistic method the lemma will follow.
Let be a multilinear polynomial that defines a constraint of . For every we define the following constraint
Observe that if , satisfy inequality (6) for every , then the lemma follows.
For every , we replace the corresponding Constraint (6) with linear constraints. For notation simplicity, let us denote the multivariate polynomial where we set and . Furthermore, let . Then, for every we create the constraint
Consider now . This can be seen as a random variable that depends on the choice of and takes values in . But recall that the ’s are sampled from using samples, and that they are mutually independent, so . Thus, we can bound the probability that a constraint (7) is not satisfied, i.e. bound the probability that , using Hoeffding’s inequality . So,
Recall, that we have constraints of the form (7). We can bound the probability that any of those constraints is violated, via the union bound. So, using (8) and the union bound, the probability that any of these constraints is violated is upper bounded by
which holds, by our choice of .
Appendix 0.D Proof of Lemma 2
To prove the lemma we will first prove the following auxiliary Lemma.
Let be a Boolean formula with one variable and one tensor-polynomial constraint of standard degree , let be a convex set, and let . If the constrained ETR problem has a solution in , then there exists a satisfiable constrained problem with variables and tensor multilinear constraints, such that every solution of in can be transformed to a solution for the constrained -ETR problem defined by and .
Assume that is a solution for . Let denote the tensor polynomial of standard degree used in . For notation simplicity, let