Near Optimal Alphabet-Soundness Tradeoff PCPs

We show that for all $\varepsilon>0$, for sufficiently large prime power $q$, for all $\delta>0$, it is NP-hard to distinguish whether a 2-Prover-1-Round projection game with alphabet size $q$ has value at least $1-\delta$, or value at most $1/q^{(1-\epsilon)}$. This establishes a nearly optimal alphabet-to-soundness tradeoff for 2-query PCPs with alphabet size $q$, improving upon a result of [Chan 2016]. Our result has the following implications: 1) Near optimal hardness for Quadratic Programming: it is NP-hard to approximate the value of a given Boolean Quadratic Program within factor $(\log n)^{(1 - o(1))}$ under quasi-polynomial time reductions. This result improves a result of [Khot-Safra 2013] and nearly matches the performance of the best known approximation algorithm [Megrestki 2001, Nemirovski-Roos-Terlaky 1999 Charikar-Wirth 2004] that achieves a factor of $O(\log n)$. 2) Bounded degree 2-CSP's: under randomized reductions, for sufficiently large $d>0$, it is NP-hard to approximate the value of 2-CSPs in which each variable appears in at most d constraints within factor $(1-o(1))d/2$ improving upon a recent result of [Lee-Manurangsi 2023]. 3) Improved hardness results for connectivity problems: using results of [Laekhanukit 2014] and [Manurangsi 2019], we deduce improved hardness results for the Rooted $k$-Connectivity Problem, the Vertex-Connectivity Survivable Network Design Problem and the Vertex-Connectivity $k$-Route Cut Problem.


INTRODUCTION
The PCP theorem is a fundamental result in theoretical computer science with many equivalent formulations [2,3,18].One of the formulations asserts that there exists > 0 such that given a satis able 3-SAT formula , it is NP-hard to nd an assignment that satis es at least (1 − ) fraction of the constraints.The PCP theorem has a myriad of applications within theoretical computer science, and of particular interest to this paper are applications of PCP to hardness of approximation.
The vast majority of hardness of approximation result are proved via reductions from the PCP theorem above.Oftentimes, to get a strong hardness of approximation result, one must rst amplify the basic PCP theorem above into a result with stronger parameters [17,21,22,26] (see [41] for a survey).To discuss these parameters, it is often convenient to view the PCP through the problem of 2-Prover-1-Round Games, which we de ne next. 1e nition 1.1.An instance Ψ of 2-Prover-1-Round Games consists of a bipartite graph = ( ∪ , ), alphabets Σ and Σ and a collection of constraints Φ = { } ∈ , which for each edge ∈ speci es a constraint map : Σ → Σ .
(2) The value of Ψ is de ned to be the maximum fraction of edges ∈ that can be satis ed by any assignment.That is, The combinatorial view of 2-Prover-1-Round Games has its origins in an equivalent, active view in terms of a game between a veri er and two all powerful provers, which is sometimes more intuitive.The veri er and the two provers have access to an instance Ψ of 2-Prover-1-Round Games, and the provers may agree beforehand on a strategy; after this period they are not allowed to communicate.The veri er then picks a random edge, = ( , ), from the 2-Prover-1-Round game, sends to the rst prover, sends to the second prover, receives a label in response from each one of them, and nally checks that the labels satisfy the constraint .If so, then the veri er accepts.It is easy to see that the value of the 2-Prover-1-Round game is equal to the acceptance probability of the veri er under the best strategy of the provers.
In the language of 2-Prover-1-Round Games, the majority of hardness of approximation results are proved by combining the basic PCP theorem [2,3,18] with Raz's parallel repetition theorem [38], which together imply the following result: Theorem 1.2.There exists > 0 such that for su ciently large , given a 2-Prover-1-Round game Ψ with alphabet size , it is NP-hard to distinguish between the following two cases: (1) YES case: val(Ψ) = 1.
For many applications, one only requires that the soundness error of the PCP is small.Namely, that val(Ψ) is arbitrarily small in the "NO case".For certain applications however, more is required: not only must the soundness error be small -but it must also be small in terms of the alphabet size.The tradeo between the soundness error of the PCP and the alphabet size of the PCP is the main focus of this paper.
With respect to this tradeo , it is clear that the best result one may hope for in Theorem 1.2 is = 1 − (1) since a random assignment to Ψ satis es, in expectation, at least 1 fraction of the constraints.In terms of results, combining the PCP theorem with Raz's parallel repetition theorem gives > 0 that is an absolute, but tiny constant.Towards a stronger tradeo , Khot and Safra [27] showed that Theorem 1.2 holds for = 1/6 with imperfect completeness (i.e., val(Ψ) ⩾ 1 − (1) instead of val(Ψ) = 1 in the "YES case").The result of Khot and Safra was improved by Chan [8], who showed (using a completely di erent set of techniques) that Theorem 1.2 holds for = 1/2 − (1), again with imperfect completeness.
In the remainder of this paper we will describe our main results and give an overview of our PCP construction.Additional details, including proofs, can be found in the full version of this paper [33].

Main Results
In this section we explain the main results of this paper.
1.1.1Near Optimal Alphabet vs Soundness Tradeo .The main result of this work improves upon all prior results, and shows that one may take = 1 − (1) in Theorem 1.2, again with imperfect completeness.Formally, we show: Theorem 1.3.For all , > 0, for su ciently large , given a 2-Prover-1-Round game Ψ, it is NP-hard to distinguish between the following two cases: Theorem 1.3 gives a near optimal tradeo between the alphabet size of a PCP and the soundness of a PCP, improving upon the result of Chan [8].Moreover, Theorem 1.3 has several applications to combinatorial optimization problems, which we discuss below.We remark that most of these applications require additional features from the instances produced in Theorem 1.3 which we omit from its formulation for the sake of clarity.For instance, one application requires a good tradeo between the size of the instance and the size of the alphabet, which our construction achieves (see the discussion following Theorem 1.4).Other applications require the underlying constraint graph to be bounded-degree bi-regular graph, which our construction also achieves, after mild modi cations detailed in [29].where , = 0 for all , and one wishes to maximize ( ) over ∈ {−1, 1} .This problem is known to have an (log ) approximation algorithm [9,31,35], and is known to be quasi-NP-hard to approximate within factor (log ) 1/6− (1) [1,27].That is, unless NP has a quasi-polynomial time algorithm, no polynomial time algorithm can approximate Quadratic Programming to within factor (log ) 1/6− (1) .As a rst application of Theorem 1.3, we improve the hardness result of Khot and Safra: Theorem 1.4.It is quasi-NP-hard to approximate Quadratic Programming to within a factor of (log ) 1− (1) .
Theorem 1.4 is proved via a connection between 2-Prover-1-Round Games and Quadratic Programming due to Arora, Berger, Hazan, Kindler, and Safra [1].This connections requires a good tradeo between the alphabet size, the soundness error, and the size of the PCP.Fortunately, the construction in Theorem 1.4 has a su ciently good tradeo between all of these parameters: letting be the size of the instance, the alphabet size can be taken to be (log ) 1− (1) and the soundness error can be taken to be (log ) −1+ (1) . 2elevance to the sliding scale conjecture: It is worth noting that using our techniques, we do not know how to achieve soundness error that is smaller than inversely poly-logarithmic in the instance size.As such, our techniques have no bearing on the sliding scale conjecture, which is concerned with getting soundness error that is inversely polynomial in the instance size.This seems to be a bottleneck of any PCP construction that is based on the covering property.In fact, assuming ETH, any quasi-polynomial PCP construction achieving soundness error, say, 1/(log ) 2 would necessarily need to have almost polynomial alphabet size (since the reduction to Quadratic Solvability would give an algorithm that runs roughly in time exponential in the alphabet size), which is the opposite of what our techniques give.With this in mind, we would like to mention a closely related, recent conjecture made in [10], which is a sort of a mixture between -to-1 games and the sliding scale conjecture.This conjecture is motivated by improved hardness results for densest -subgraph style problems, and focuses on the relation between the instance size and the soundness error (allowing the alphabet to be quite large).It may be possible that the ideas from the current paper can help make progress towards this conjecture.
1.1.3Application: NP-hardness of Approximating Bounded Degree 2-CSPs.Theorem 1.3 has an application to the hardness of approximating the value of 2-CSPs with bounded degree, as we explain next.
An instance Ψ of 2-CSP, say Ψ = ( , , Σ), consists of a set of variables , a set of constraints and an alphabet Σ.Each constraint in has the form ( , ) = 1 where : Σ × Σ → {0, 1} is a predicate (which may be di erent in distinct constraints).The degree of the instance Ψ is de ned to be the maximum, over variables ∈ , of the number of constraints that appears in.The goal is to nd an assignment : → Σ that satis es as many of the constraints as possible.
There is a simple +1 2 approximation algorithm for the 2-CSP problem for instances with degree at most .Lee and Manurangsi proved a nearly matching 1  2 − (1) hardness of approximation result assuming the Unique-Games Conjecture [29].Unconditionally, they show the problem to be NP-hard to approximate within factor 1  3 − (1) under randomized reductions.Using the ideas of Lee and Manurangsi, our main result implies a nearly matching NP-hardness result for bounded degree 2-CSPs: Theorem 1.5.For all > 0, for su ciently large , approximating the value of 2-CSPs with degree at most within factor 1  2 − is NP-hard under randomized reductions.
As in [29], Theorem 1.5 has a further application to nding independent sets in claw free graphs.A -claw 1, is the ( + 1) vertex graph with a center vertex which is connected to all othervertices and has no other edges; a graph is said to be -claw free if does not contain an induced -claw graph.There is a polynomial time approximation algorithm for approximating the size of the largest independent set in a given -claw free graph within factor 2 [4,40], and a quasi-polynomial time approximation algorithm within factor 1  3 + (1) [11].As in [29], using ideas from [14] Theorem 1.5 implies that for all > 0, for su ciently large , it is NP-hard (under randomized reductions) to approximate the size of the largest independent set in a given -claw free graph within factor 1  4 + .This improves upon the result of [29] who showed that the same result holds assuming the Unique-Games Conjecture.
1.1.4Application: NP-hardness of Approximating Connectivity Problems.Using ideas of Laekhanukit [28] and the improvements by Manurangsi [30], Theorem 1.3 implies improved hardness of approximation results for several graph connectivitiy problems.More speci cally, Theorem 1.3 combined with the results of [30] implies improvements to each one of the results outlined in table 1 in [28] by a factor of 2 in the exponent -with the exception of Rooted--Connectivity on directed graphs where a factor of 2 improvement is already implied by [30].We brie y discuss the Rooted -Connectivity Problem, but defer the reader to [28] for a detailed discussion of the remaining graph connectivity problems.
In the Rooted -Connectivity problem there is a graph = ( , ), edge costs : → R, a root vertex ∈ and a set of terminals ⊆ \ { }.The goal is to nd a sub-graph ′ of smallest cost that for each ∈ , has at least vertex disjoint paths from to .The problem admits | | trivial approximation algorithm (by applying minimum cost -ow algorithm for each vertex in ), as well as an ( log ) approximation algorithm [36].
Using the ideas of [28], Theorem 1.3 implies the following improved hardness of approximation results: Theorem 1.6.For all > 0, for su ciently large it is NP-hard to approximate the Rooted--Connectivity problem on undirected graphs to within a factor of 1/5− , the Vertex-Connectivity Survivable Network Design Problem with connectivity parameters at most to within a factor of 1/3− , and the Vertex-Connectivity -Route Cut Problem to within a factor of 1/3− .We remark that in [7], a weaker form of hardness for the Vertex-Connectivity Survivable Network problem is proved.More precisely, they show an Ω( 1/3 /log ) integrality gap for the set-pair relaxation of the problem.Our hardness result of 1/3− improves upon it, showing that (unless P=NP) no relaxation can yield a better than 1/3− factor approximation algorithm.

PRELIMINARIES
In this section we will describe some preliminary de nitions and results.We rst present the Grassmann graph and some Fourier analytic tools that are used in our analysis.We then state some hardness results regarding 3Linwhich will form the starting point of our PCP construction.

The Grassmann Graph
Throughout this section, we x parameters , ℓ with 1 ≪ ℓ ≪ , and a prime power .The Grassmann graph Grass ( , ℓ) is de ned as follows.
• The vertex set corresponds to the set of ℓ-dimensional subspaces ⊆ F .
We also write Grass ( , ℓ) to denote the Grassmann graph on ℓdimensional subspaces , where is some large linear subspace.Finally, we denote by 2 (Grass ( , ℓ)) the set of complex valued functions : Zoom ins and Zoom outs.A feature of the Grassmann graph is that it contains many copies of lower dimensional Grassmann graphs as induced subgraphs.These subgraphs are precisely the zoom-ins and and zoom-outs referred to in the introduction, and they play a large part in the analysis of our inner PCP and nal PCP.For subspaces ⊆ ⊆ F , let We refer to as a zoom-in and as a zoom-out.When = F , Zoom[ , ] is the zoom-in on , and when = {0}, Zoom[ , ] is the zoom-out on .
2.1.1Pseudo-randomness over the Grassmann graph.One notion that will be important to us is ( , )-pseudo-randomness, which measures how much can deviate from its expectation on a zoomin/zoom-out restrictions of "size ".For all of our applications, and will both be indicator functions of some sets of vertices, so it will be helpful to think of this case for the remainder of the section. 3Let ( ) = E ∈Grass ( ,ℓ ) [ ( )] (for indicator functions, this is simply the measure of the indicated set).For subspaces ⊆ ⊆ F , de ne De nition 2.1.We say that a Boolean function : We will often say that a set ⊆ Grass ( , ℓ) is ( , )-pseudorandom if its indicator function is.Because the Grassmann graph is not a small-set expanders, there are small sets in it that do not look "random" with respect to some combinatorial counting measures (such as edges between sets, expansion and so on).Intuitively, a small set which is highly pseudo-random will exhibit random-like structure with respect to several combinatorial measures of interest, and the two lemmas below are instantiations of it required in our proof.The proof proceed by reducing them to similar statements about the Bi-linear scheme, which can then be proved directed by appealing to global hypercontractivity results of [15,16].
For the analysis of the inner PCP, we require the following lemma, which bounds the number of edges between a subset, L, of Grass ( , 2ℓ), and Grass ( , and suppose that is ( , ) pseudo-random.Then for all ⩾ 4 that are powers of 2, ( −1)/ 2 /(2 −1) + − ℓ .

Hardness of 3LIN
In this section we cite several hardness of approximation results for the problem of solving linear equations over nite elds, which are the starting point of our reduction.We begin by de ning the 3Lin and the Gap3Lin problem.
De nition 2.3.For a prime power , an instance of 3Lin is ( , Eq) which consists of a set of variables and a set of linear equations Eq over F .Each equation in Eq depends on exactly three variables in , each variable appears in at most 10 equations, and any two distinct equations in Eq share at most a single variable.
The goal in the 3Lin problem is to nd an assignment : → F satisfying as many of the equations in as possible.The maximum fraction of equations that can be satis ed is called the value of the instance.We remark that usually in the literature, the condition that two equations in share at most a single variable is not included in the de nition of 3Lin, as well the the bound on the number of occurences of each variable.
For 0 < < ⩽ 1, the problem Gap3Lin[ , ] is the promise problem wherein the input is an instance ( , ) of 3Lin promised to either have value at least or at most , and the goal is to distinguish between these two cases.The problem Gap3Lin[ , ] with various settings of and will be the starting point for our reductions.
To prove Theorem 1.3, we shall use the classical result of Håstad [22].This result says that for general 3Lin instances (i.e., without the additional condition that two equations share at most a single variable), the problem Gap3Lin[1 − , 1/ + ] is NP-hard for all constant ∈ N and > 0. This result implies the following theorem by elementary reductions: Theorem 2.1.There exists < 1 such that for every constant > 0 and prime , Gap3Lin [1 − , ] is NP-hard.
To prove Theorem 1.4 we will need a hardness result for 3Lin with completeness close to 1, and we will use a hardness result of Khot and Ponnuswami [26].Once again, their result does not immediately guarantee the fact that any two equations share at most a single variable, however once again this property may be achieved by an elementary reduction.
Theorem 2.2.There is a reduction from SAT with size to an instance of Gap3Lin[1− , 1− ] of size over a eld F of characteristic 2, where, • Both and the running time of the reduction are bounded by

THE PCP CONSTRUCTION
Theorem 1.3 is proved by composing an inner PCP and an outer PCP.Both of these components incorporate ideas from the proof of the 2-to-1 Games Theorem.The outer PCP is constructed using smooth parallel repetition [24,27] while the inner PCP is based on the Grassmann graph [12,13,24,25].
The novelty in this current paper, in terms of techniques, is twofold.First, we must consider a Grassmann test in a di erent regime of parameters (as otherwise we would not be able to get a good alphabet to soundness tradeo ) and in a regime of much lower soundness error.These di erences complicate matters considerably.Second, our soundness analysis is more involved than that of the 2-to-1-Games Theorem.As is the case in [12,13,24,25], we too use global hyperconractivity, but we do so more extensively.We also require quantitatively stronger versions of global hypercontractivity over the Grasssmann graph which are due to [16].In addition, our analysis incorporates ideas from the plane versus plane test and direct product testing [23,32,39], from classical PCP theory [27], as well as from error correcting codes [19].All of these tools are necessary to prove our main technical statement -Lemma 3.1 below -which is a combinatorial statement that may be of independent interest.
We now elaborate on each one of the components separately.

The Inner PCP
Our Inner PCP is based on the subspace vs subspace low degree test.Below, we rst give a general overview of the objective in lowdegree testing.We then discuss the traditional notion of soundness as well as a non-traditional notion of soundness for low-degree tests.Finally, we explain the low-degree test used in this paper, the notion of soundness that we need from it, and the way that this notion of soundness is used.
Low degree tests in PCPs.Low degree tests have been have a vital component in PCPs since their inception, and much attention has been devoted to improving their various parameters.The goal in low-degree testing is to encode a low-degree function : F → F via a table (or a few tables) of values, in a way that allows for local testing.Traditionally, one picks a parameter ℓ ∈ N (which is thought of as a constant and is most often just 2) and encodes the function by the table of restrictions of to ℓ-dimensional a ne subspaces of F .For the case ℓ = 2, the test associated with this encoding is known as the Plane vs Plane test [39].The Plane vs Plane test proceeds by picking two planes 1 , 2 intersecting on a line, and then checking that [ 1 ] and [ 2 ] agree on 1 ∩ 2 .It is easy to see that the test has perfect completeness, namely that a valid table of restrictions passes the test with probability 1.In the other direction, the soundness error of the test -which is a converse type statement -is much less clear (and is crucial towards applications in PCP).In the context of the Plane vs Plane test, it is know that if a table , that assigns to each plane a degree function, passes the Plane vs Plane test with probability ⩾ − (where > 0 is a small absolute constant), then there is a degree function such that [ ] ≡ | on at least Ω( ) fraction of the planes.
Nailing down the value of the constant for which soundness holds is an interesting open problem which is related to soundness vs alphabet size vs instance size tradeo in PCPs [5,32,34].Currently, the best known analysis for the Plane vs Plane test [34] shows that one may take = 1/8.Better analysis is known for higher dimensional encoding [5,32], and for the 3-dimensional version of it a near optimal soundness result is known [32].
Low degree tests in this paper.In the context of the current paper, we wish to encode linear functions : F → F , and we do so by the subspaces encoding.Speci cally, we set integer parameters ℓ 1 ⩾ ℓ 2 , and encode the function using the table 1 of the restrictions of to all ℓ 1 -dimensional linear subspaces of F , and the table 2 of the restrictions of to all ℓ 2 -dimensional linear subspaces of F .The test we consider is the natural inclusion test: (1) Sample a random ℓ 1 -dimensional subspace 1 ⊆ F and a random ℓ 2 -dimensional subspace and accept if they agree on 2 .
As is often the case, the completeness of the test -namely the fact that valid tables 1 , 2 pass the test with probability 1 -is clear.The question of most interest then is with regards to the soundness of the test.Namely, what is the smallest such that any two tables 1 and 2 that assign linear functions to subspaces and pass the test with probability , must necessarily "come from" a legitimate linear function ?
Traditional notion of soundness.As the alphabet vs soundness tradeo is key to the discussion herein, we begin by remarking that the alphabet size of the above encoding is ℓ 1 + ℓ 2 = Θ( ℓ 1 ) (since there are ℓ distinct linear functions on a linear space of dimension ℓ over F ). Thus, ideally we would like to show that the soundness error of the above test is − (1− (1) )ℓ 1 .Alas, this is false.Indeed, it turns out that one may construct assignments that pass the test with probability at least Ω(max( −ℓ 2 , ℓ 2 −ℓ 1 )) that do not have signi cant correlation with any linear function : (1) Taking ) many possible 's, 1 has Θ(1) many possible 's and furthermore there is at least one that is valid for both of them.With probability Ω( ℓ 2 −ℓ 1 ) this common is chosen for both 1 and 2 , and in this case, the test on ( 1 , 2 ) passes.It follows that, in expectation, 1 , 2 pass the test with probability Ω( ℓ 2 −ℓ 1 ).In light of the above, it makes sense that the best possible alphabet vs soundness tradeo we may achieve with the subspace encoding is by taking ℓ 2 = ℓ 1 /2.Such a setting of the parameters would give alphabet size = ℓ 1 and (possibly) soundness error Θ(1/ √ ).There are several issues with this setting however.First, this tradeo is not good enough for our purposes (which already rules out this setting of parameters).Second, we do not know how to prove that the soundness error of the test is Θ(1/ √ ) (the best we can do is quadratically o and is Θ(1/ 1/4 )).To address both of these issues, we must venture beyond the traditional notion of soundness.
Non-traditional notion of soundness.The above test was rst considered in the context of the 2-to-1 Games Theorem, wherein one takes = 2 and ℓ 2 = ℓ 1 − 1.In this setting, the test is not sound in the traditional sense; instead, the test is shown to satisfy a nonstandard notion of soundness, which nevertheless is su cient for the purposes of constructing a PCP.More speci cally, in [25] it is proved that for all > 0 there is ∈ N such that for su ciently large ℓ and for tables 1 , 2 as above, there are subspaces ⊆ ⊆ F with dim( ) + codim( ) ⩽ and a linear function : → F such that Pr We refer to the set as the zoom in of and zoom out of .While this result is good for the purposes of 2-to-1 Games, the dependency between ℓ and (and thus, between the soundness and the alphabet size) is still not good enough for us.
Our low-degree test.It turns out that the proper setting of parameters for us is ℓ 2 = (1 − )ℓ 1 where > 0 is a small constant.With these parameters, we are able to show that for ⩾ − (1− ′ )ℓ 1 (where ′ = ′ ( ) > 0 is a vanishing function of ), if 1 , 2 pass the test with probability at least , then there are subspaces ⊆ with dim( ) + codim( ) ⩽ = ( ) ∈ N, and a linear function This result is obtained from Lemma 2.2, which in turn relies on [16].
Working in the very small soundness regime of ⩾ − (1− ′ )ℓ 1 entails with it many challenges, however.First, dealing with such small soundness requires us to use a strengthening of the global hypercontractivity result of [25] in the form of an optimal level inequality due to Evra, Kindler and Lifshitz [16].Second, in the context of [25], ′ could be any function of (and indeed it ends up being a polynomial function of ).In the context of the current paper, it is crucial that ′ = 1+ (1) , as opposed to, say, ′ = 1.1 .The reason is that, as we are dealing with very small , the result would be trivial for ′ = 1.1 and not useful towards the analysis of the PCP (as then ′ would be below the threshold −ℓ 1 which represents the agreement a random linear function has with 1 ).

Getting List Decoding Bounds
As is usually the case in PCP reductions, we require a list decoding version for our low-degree test.Indeed, using a standard argument we are able to show that in the setting that ℓ 2 = (1 − )ℓ 1 and ⩾ (1− ′ )ℓ 1 , there is = ( , ′ ) ∈ N such that for at least −Θ(ℓ 1 ) fraction of subspaces ⊆ F of dimension , there exists a subspace with co-dimension at most and ⊆ ⊆ F , as well as a linear function : → F , such that This list decoding version theorem alone is not enough.In our PCP construction, we compose the inner PCP with an outer PCP (that we describe below), and analyzing the composition requires decoding global linear functions (from a list decoding version theorem as above) in a coordinated manner between two non communicating parties.Often times, the number of possible global functions that may be decoded is constant, in which case randomly sampling one among them often works.This is not the case for us, though: if ( , ) and ( ′ , ′ ) are distinct zoom-in and zoom-out pairs for which there are linear functions , and ′ , ′ satisfying (1), then the functions , and ′ , ′ could be completely di erent.Thus, to achieve a coordinated decoding procedure, we must: (1) Facilitate a way for the two parties to agree on a zoom-in and zoom-out pair ( , ) with noticeable probability.(2) Show that for each ( , ) there are at most poly(1/ ) functions , for which The second item is precisely the reason we need ′ to be 1+ (1) ; any worse dependency, such as ′ = 1.1 would lead to the second item being false.We also remark that the number of functions being poly(1/ ) is important to us as well.There is some slack in this bound, but a weak quantitative bound such as exp(exp(1/ )) would have been insu cient for some of our applications.Luckily, such bounds can be deduced from [19] for the case of linear functions. 4e now move onto the rst item, in which we must facilitate a way for two non-communicating parties to agree on a zoom-in and zoom-out pair ( , ).It turns out that agreeing on the zoom-in can be delegated to the outer PCP, and we can construct a sound outer PCP game in which the two parties are provided with a coordinated zoom-in .This works because in our list decoding theorem, the fraction of zoom-ins that work is signi cant.Coordinating zoom-outs is more di cult, and this is where much of the novelty in our analysis lies.

Coordinating Zoom-outs
For the sake of simplicity and to focus on the main ideas, we ignore zoom-ins for now and assume that the list decoding statement holds with no .Thus, the list decoding theorem asserts that there exists a zoom-out of constant co-dimension on which there is a global linear function.However, there could be many such zoomouts , say 1 , . . ., and say all of them were of co-dimension . If the number were su ciently large -say at least −poly(ℓ 1 ) fraction of all co-dimension subspaces -then we would have been able to coordinate them in the same way as we coordinate zoom-ins.If the number were su ciently small -say = poly(ℓ 1 ) , then randomly guessing a zoom-out would work well enough.The main issue is that the number is intermediate, say This issue had already appeared in [12,24].Therein, this issue is resolved by showing that if there are at least ⩾ 100ℓ 2 1 zoom-outs 1 , . . ., of co-dimension , and linear functions 1 , . . ., on 1 , . . ., respectively such that for all , then there exists a zoom out of co-dimension strictly less than and a linear function : Thus, if there are too many zoom-outs of a certain co-dimension, then there is necessarily a zoom-out of smaller co-dimension that also works.In that case, the parties could go up to that co-dimension.This result is not good enough for us, due to the polynomial gap between the agreement between and 's and and the agreement between an .Indeed, in our range of parameters, ′12 will be below the trivial threshold −ℓ 1 which is the agreement a random linear function has with , and therefore the promise on the function above is meaningless.
We resolve this issue by showing a stronger, essentially optimal version of the above assertion still holds.Formally, we prove: Lemma 3.1.For all > 0, ∈ N there is > 1 such that the following holds for ′ ⩾ (1− )ℓ 1 .Suppose that is a table that assigns to each subspace of dimension ℓ 1 a linear function, and suppose that there are at least ⩾ ℓ 1 subspaces 1 , . . ., of co-dimension and linear functions : for all = 1, . . ., .Then, there exists a zoom-out of co-dimension strictly smaller than and a linear function : We remark that our proof of Lemma 3.1 is very di erent from the arguments in [12] and is signi cantly more involved.Our proof uses tools from [12,24], tools from the analysis of the classical Plane vs Plane and direct product testing [23,32,39], global hypercontractivity [16] as well as Fourier analysis over the Grassmann graph.

The Outer PCP
Our outer PCP game is the outer PCP of [12,24], which is a smooth parallel repetition of the equation versus variables game of Hastad [22] (or of [26] for the application to Quadratic Programming).As in there, we equip this game with the "advice" feature to facilitate zoom-in coordination (as discussed above).For the sake of completeness we elaborate on the construction of the outer PCP below.
We start with an instance of 3-Lin that has a large gap between the soundness and completeness.Namely, we start with an instance ( , ) of linear equations over F in which each equation has the form 1 + 2 + 3 = .It is known [22] that for all > 0, it is NP-hard to distinguish between the following two cases: (1) YES case: val( , ) ⩾ 1 − .(2) NO case: val( , ) ⩽ 1.1 .
(2) With probability , the veri er sets = { 1 , 2 , 3 }, chooses a set consisting of a single variable ⊆ uniformly at random.The veri er picks 1 , . . ., ∈ F uniformly and independently and appends to each the value 0 in the coordinates of \ to get 1 , . . ., .
After that, the veri er sends and 1 , . . ., to the rst prover and and 1 , . . ., to the second prover.The veri er expects to get from them F assignments to the variables in and in , and accepts if and only if these assignments are consistent, and furthermore the assignment to satis es the equation .
The game Ψ ⊗ is our outer PCP game.Remark 3.2.We remark that in the case of the Quadratic Programming application, we require a hardness result in which the completeness is very close to 1 in the form of Theorem 2.2.The di erences between the reduction in that case and the reduction presented above are mostly minor, and amount to picking the parameters a bit di erently.There is one signi cant di erence in the analysis; we require a much sharper form of the "covering property" used in [12,24], as elaborated on in Section 3.6

Composing the Outer PCP and the Inner PCP Game
To compose the outer and inner PCPs, we take the outer PCP game, only keep the questions to the rst prover, and consider an induced 2-Prover-1-Round game on it.The alphabet is F 3 : given a question , the alphabet is the set of F assignment to the variables of .There is a constraint between and ′ if there is a question to the second prover such that ⊆ ∩ ′ .Denoting the assignments to and ′ by and ′ , the constraint between and ′ is that satis es all of the equations that form , ′ satis es all of the equations that form ′ , and , ′ agree on ∩ ′ .
The composition amounts to replacing each question with a copy of our inner PCP.Namely, we identify between the question and the space F , and then replace by a copy of the ℓ 2 , ℓ 1 sub-spaces graph of F .The answer is naturally identi ed with the linear function ( ) = ⟨ , ⟩, which is then encoded by the sub-spaces encoding via tables of assignments 1, and 2, .
The constraints of the composed PCP must check two things: (1) side conditions: the encoded vector satis es the equations of , and (2) consistency: and ′ agree on ∩ ′ .The rst set of constraints is addressed by the folding technique, which we omit from this discussion.The second set of constraints is addressed by the ℓ 1 vs ℓ 2 subspace test, except that we have to modify it so that it works across blocks and ′ .This completes the description of the composition step of the outer PCP and the inner PCP, and thereby the description of our reduction.
Let us brie y describe the setting of parameters used to obtain Theorem 1.3.After xing the , therein, we may take = 2, choose ′ su ciently small according to and , set ℓ 2 = (1 − ′ )ℓ 1 , and nally take ℓ 1 su ciently large.We must also choose and carefully to satisfy the covering property and completeness of the composed PCP, but omit further details from the current discussion.Altogether this yields alphabet size ℓ 1 and soundness − (1− )ℓ 1 .We remark that the same tradeo can be obtained with larger settings of and this is indeed required for the application to hardness of approximating quadratic programming in Theorem 1.4.

The Covering Property
We end by brie y discussing the covering property.The covering property is an important feature of our outer PCP construction which enables the composition step to go through.The covering property rst appeared in [27] and later more extensively in the context of the 2-to-1 Games [12,24].To discuss the covering property, let ∈ N be thought of as large, let ∈ (0, 1) be thought of as −0.99 and consider sets 1 , . . ., consisting of distinct element, each has size 3 (in our context, will be the set of variables in the th equation the veri er chose).Let = 1 ∪ . . .∪ , and consider the following two distributions over tuples in F : (1) Sample 1 , . . ., ℓ ∈ F uniformly.
In [24] it is shown that the two distributions above are 3ℓ √ close in statistical distance, which is good enough for the purposes of Theorem 1.3.However, this is not good enough for Theorem 1.4. 5Carrying out a di erent analysis, we are able to show that the two distributions are close with better parameters and in a stronger sense: there exists a set of ℓ tuples which has negligible measure in both distributions, such that each tuple not in is assigned the same probability under the two distribution up to factor (1 + (1)).We are able to prove this statement provided that is only slightly larger than 2ℓ .
The issue with the above two distributions is that they are actually far from each other if, say, = 1.9ℓ .To see that, one can notice that the expected number of 's such that each one of 1 , . . ., ℓ has the form ( , 0, 0) ∈ F 3 on coordinates corresponding to is very di erent.In the rst distribution, this expectation is Θ( −2ℓ ) which is less than 1, whereas in the second distribution it is at least ⩾ 0.01 .To resolve this issue and to obtain nearly tight hardness in the Quadratic Programming application, we have to modify the distributions in the covering property so that (a) they will be close even if = 1.01ℓ , and (b) we can still use these distributions in the composition step in our analysis of the PCP construction.Indeed, this is the route we take, and the two distributions we use are de ned as follows: (1) Sample 1 , . . ., ℓ ∈ F uniformly.
(2) For each independently, take = with probability 1 − and otherwise take ⊆ randomly of size 1, then set where , are independent random elements from F .Output 1 , . . ., ℓ .We show that for a suitable choice of and , these distributions are close even in the case that = 1.01ℓ . 6Indeed, as a sanity check one could count the expected number of appearances of blocks of the form (0, , 0) ∈ F 3 and see they are very close ( −2ℓ versus (1 − ) −2ℓ + −ℓ ).In this setting of parameters, is roughly equal to the alphabet size -which can be made to be equal (log ) 1− (1) under quasi-polynomial time reductions -it is su cient to get the result of Theorem 1.4.Remark 3.3.We remark that a tight covering property is crucial for obtaining the tight hardness of approximation factor in Theorem 1.4.In the reduction from 2-Prover-1-Round games to Quadratic Programs, which is due to [1], the size of the resulting instance is exponential in the alphabet size and the soundness error remains roughly the same.In our case the alphabet size is roughly hence the instance size is dominated by = 2 Θ( 1+ (1) ) .If our analysis required = ℓ , then even showing an optimal soundness of − (1− (1) )ℓ for the 2-Prover-1-Round game would only yield a factor of (log ) 1/ − (1) hardness for quadratic programming.

1. 1 . 2
Application: NP-Hardness of Approximating adratic Programs.Theorem 1.3 has an application to the hardness of approximating the value of Boolean Quadratic Programming, as we explain next.An instance of Quadratic programming consists of a quadratic form ( ) = , =1 ,