On the NP-Hardness of Approximating Ordering-Constraint Satisfaction Problems

www.theoryofcomputing.org

S

: APPROX-RANDOM 2013

On the

NP-Hardness of Approximating

Ordering-Constraint Satisfaction Problems

Per Austrin

Rajsekar Manokaran

Cenny Wenner

Received October 2, 2013; Revised November 9, 2014; Published June 18, 2015

Abstract: We show improvedNP-hardness of approximatingOrdering-Constraint Satis-faction Problems(OCSPs). For the two most well-studied OCSPs, MAXIMUMACYCLIC SUBGRAPHand MAXIMUMBETWEENNESS, we proveNP-hard approximation factors of 14/15+ε and 1/2+ε, respectively. When it is hard to approximate an OCSP by a constant better than taking a uniform random ordering, then the OCSP is said to beapproximation resistant.We show that the MAXIMUMNon-BETWEENNESSPROBLEMis approximation resistant and that there are width-mapproximation-resistant OCSPs accepting only a fraction 1/(m/2)! of assignments. These results provide the first examples of approximation-resistant OCSPs subject only toP6=NP.

ACM Classification:F.2.2

AMS Classification:68Q17, 68W25

Key words and phrases: acyclic subgraph, APPROX, approximation, approximation resistance, be-tweenness, constraint satisfaction, CSPs, feedback arc set, hypercontractivity, NP-completeness, orderings, PCP, probabilistically checkable proofs

Anextended abstractof this paper appeared in the 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2013) [4].

∗_{Supported by Swedish Research Council Grant 621-2012-4546.} †_{Supported by ERC Advanced Grant 226203.}

1

Introduction

x z y

a

b

Figure 1: An MAS in-stance with value 5/6. We study theNP-hardness of approximating a rich class of optimization problems

known asOrdering-Constraint Satisfaction Problems(OCSPs). An instance of an OCSP is given by a setXof variables and a setCoflocal ordering constraints

where each constraint is specified by a set of permitted permutations on a subset of the variables. The objective is to find an ordering ofXmaximizing the fraction of constraints satisfied by the induced local permutations.

A simple example of an OCSP is the problem MAXIMUMACYCLICSUB -GRAPH(MAS) where one is given a directed graphG= (V,A)with the task of finding an acyclic subgraph ofGcontaining as many arcs as possible. Phrased as an OCSP,V is the set of variables and each arcu→vis a constraint “u≺v” dictating thatushould precedev. The maximum fraction of constraints simulta-neously satisfiable by an ordering ofV is then precisely the fraction of edges in a

largest acyclic subgraph. Since each constraint involves exactly two variables, MAS is an OCSP ofwidth

two. Another example of an OCSP is the MAXIMUMBETWEENNESS(MAXBTW) problem [12]. In this width-three OCSP, a constraint on a triplet of variables(x,y,z)is satisfied by the local orderingsx≺z≺y

andy≺z≺x– in other words,zhas to be betweenxandy, giving rise to the name of the problem. Determining the optimal value of an MAS instance isNP-hard and one turns to approximations. An algorithm is called ac-approximation if, when applied to an instanceI, the algorithm is guaranteed to produce an orderingOsatisfying a fraction of constraints within a factorc of the optimum, i. e., val(O;I)≥c·val(I). As in the case of classical constraint satisfaction, every OCSP admits a naive approximation algorithm which picks an ordering ofXuniformly at random without even looking at the constraints. For example, this algorithm yields an 1/2-approximation in expectation for MAS.

Problem Approx. factor UG-inapprox. NP-inapprox. This work MAS 1/2+Ω(1/logn)[8] 1/2+ε[14] 65/66+ε[19] 14/15+ε MAXBTW 1/3 1/3+ε[7] 47/48+ε[9] 1/2+ε

MAXNBTW 2/3 2/3+ε[7] - 2/3+ε

m-OCSP 1/m! 1/m!+ε [13] - 1/bm/2c!+ε

Table 1:Prior results and improvements.

1.1 Results

In this work we obtain improvedNP-hardness of approximating various OCSPs. While a complete characterization such as in the UG regime still eludes us, our results improve the knowledge of what we believe are four important flavors of OCSPs; seeTable 1for a summary of the present state of affairs.

We address the two most studied OCSPs: MAS and MAX BTW. For MAS, we show a factor (14/15+ε)-inapproximability improving the factor from 65/66+ε [19]. For MAXBTW, we show a factor(1/2+ε)-inapproximability improving from 47/48+ε[9].

Theorem 1.1. For everyε>0, it isNP-hard to distinguishMASinstances with value at least15/18−ε

from instances with value at most14/18+ε.

Theorem 1.2. For everyε>0, it isNP-hard to distinguishMAX BTWinstances with value at least 1−εfrom instances with value at most1/2+ε.

The above two results are inferior to what is known assuming the UGC and in particular do not prove approximation resistance. We introduce the MAXIMUM NON-BETWEENNESS(MAXNBTW) problem which accepts thecomplementof the predicate in MAXBTW. This predicate accepts four of the six permutations on three elements and thus a random ordering satisfies two thirds of the constraints in expectation. We show that this is optimal up to smaller-order terms.

Theorem 1.3. For everyε>0, it isNP-hard to distinguishMAXNBTWinstances with value at least

1−εfrom instances with value at most2/3+ε.

Finally, we address the approximability of a generic width-mOCSP as a function of the widthm. In the CSP world, the generic version is calledm-CSP and we call the ordering versionm-OCSP. We devise a simple predicate, “2t-Same Order” 2t-SO, onm=2tvariables that is satisfied only if the firstt

elements are relatively ordered exactly as the lasttelements. A random ordering satisfies only a fraction 1/t! of the constraints and we prove that this is essentially optimal, implying a(1/bm/2c!+ε)-factor inapproximability ofm-OCSP.

Theorem 1.4. For everyε>0and integer m≥2, it isNP-hard to distinguish m-OCSP instances with

value at least1−ε from value at most1/bm/2c!+ε.

1.2 Proof overviews

gadgets, also known as long-code tests. We describe these reductions in the context of MAXNBTW to highlight the new techniques in this paper.

The reduction produces an instanceIof MAXNBTW from an instanceLof LC such thatval(I)> 1−ε ifval(L) =1 while val(I)<2/3+ε ifval(L)≤η for some η=η(ε). By the PCP Theorem and the Parallel Repetition Theorem [3,2, 20], it isNP-hard to distinguish betweenval(L) =1 and val(L)≤ηfor every constantη>0 and thus we obtain the result inTheorem 1.3. The core component in this paradigm is the design of a dictatorship test: a MAXNBTW instance on[q]L∪[q]R, for integersq

and label setsLandR. Letπ be a mapR→L. Each constraint is a tuple(~x,~y,~z)where~x∈[q]L_{, while}

~y,~z∈[q]R. The distribution of tuples is obtained as follows. First, pick~x, and~yuniformly at random from [q]L, and[q]R. Setzj=yj+xπ(j)modq. Finally, add noise by independently replacing each coordinate

xi,yj andzj with a uniformly random element from[q]with probabilityγ.

This test instance has canonical assignments that satisfy almost all the constraints. These are obtained by picking an arbitrary j∈[R], and partitioning the variables intoqsetsS0, . . .Sq−1where

St ={~x∈[q]R|xπ(j)=t} ∪ {~y∈[q]L|yj=t}.

If a constraint(~x,~y,~z) is so that~x∈St,~y∈Su then~z∈/Sv for anyv∈ {t+1, . . . ,u−1}except with

probabilityO(γ). This is because(a+b)modqis never strictly betweenaandb. Further, the probability that any two of~x,~y, and~zfall in the same setSiis simply the probability that any two ofxπ(j),yj, andzj

are assigned the same value, which is at mostO(1/q). Thus, ordering the variables such thatS₀≺S₁≺ . . .≺Sq−1with an arbitrary ordering of the variables within a set satisfies a fraction 1−O(1/q)−O(γ) constraints.

The proof ofTheorem 1.3requires a partial converse of the above: every ordering that satisfies more than a fraction 2/3+ε of the constraints is more-or-less an ordering that depends on a few coordinates j as above. This proof involves three steps. First, we show that there is aΓ=Γ(q,γ,β)such that every orderingOof[q]Land[q]Rcan be broken intoΓsetsS0, . . . ,SΓ−1such that one achieves expected value at leastval(O)−β for arbitrarily smallβ by ordering the setsS0≺ · · · ≺SΓ−1and within each set ordering elements arbitrarily. The proof of this “bucketing” uses hypercontractivity of noised functions from a finite domain. We note that a related bucketing argument is used in proving inapproximability of OCSPs assuming the UGC [14,13]. Somewhat similar to our analysis, they “round” table values toqbuckets of equal size and proceed to analyze indicator functions for ranges of order assignments. In particular, the UG-based analysis only needs to consider multiple queries to, and bucketing of, individual tables. For our construction, we have to consider the simultaneous bucketing of two tables. Merely “rounding” the tables to a set[q]does not suffice. Rather, we consider having q equal-sized buckets for each table and analyze functions which do not differentiate values within buckets. More precisely, we introduce a “bucketed payoff” which, given a tuple of bucket indices, samples a random element from the respective indexed buckets of the tables and evaluates the predicate for the resulting tuple. With this setup, we can analyze functions of bounded range, yet the bucketing preserves arbitrarily well the value of an instance even when the tables have influential coordinates.

“decoupled” from(~y,~z). We note that the marginal distribution of the tuple(~y,~z)is already symmetric with respect to swaps:P(~y=y,~z=z) =P(~y=z,~z=y). In order to prove approximation resistance, we combine three of these dictatorship tests: the jth variant has~xas the jth component of the triple. We show that the combined instance is symmetric with respect to every swap up to an errorO(ε)unless val(L)>η. This implies that the instance has value at most 2/3+O(ε)hence proving approximation resistance of MAXNBTW.

For MAX BTW and MAX 2t-SO, we do not require the final symmetrization and instead use a dictatorship test based on a different distribution. Finally, the reduction to MAS is a simple gadget reduction from MAX NBTW. For hardness results of width-two predicates, such gadget reductions presently dominate the scene of classical CSPs and also designates the state of affairs for MAS. As an example, the best-knownNP-hard approximation hardness of 16/17+ε for MAXCUTis via a gadget reduction from MAX3-LIN-2 [16,23]. The preceding best approximation hardness of MAS was also via a gadget reduction from MAX3-LIN-2 [19], although with the significantly smaller hardness constant 65/66+ε. By reducing from a problem more similar to MAS, namely MAXNBTW, we improve to the approximation hardness to 14/15+ε. The gadget in question is simple and was given inSection 1.

Organization. Section 2sets up the notation used in the rest of the article. Section 3gives a general hardness result based on test distributions, although the proofs are deferred toSection 5. InSection 4, we introduce distributions tailored to the studied predicates and invoke the construction fromSection 3to prove three of the four approximation-hardness results; the fourth result is proven in the same section via a gadget reduction.

Acknowledgments. We would like to thank Johan Håstad for suggesting the topic and for numerous helpful discussions regarding the same, as well as the anonymous referees, and the editors of the publishing issue, who assisted in improving the legibility of the work.

2

Preliminaries

We denote by[n]the integer interval{0, . . . ,n−1}and by[n]₁the interval{1, . . . ,n}. Given a tuple of realsx∈_Rm_{, we write~}

σ(x)∈Smfor the natural-order permutation on{1, . . . ,m}induced byx. We also

denote byx−i the tuple(x1, . . . ,xi−1,xi+1, . . . ,xm). For a distributionDoverΩ1× · · · ×Ωm, we useD≤t

andD>t to denote the projection to coordinates up totand to the remaining coordinates, respectively.

2.1 Ordering-constraint satisfaction problems

We are concerned with predicatesP:Sm→[0,1]on the symmetric groupSm. Such a predicate specifies a

width-mOCSP written as OCSP(P). An instanceIof OCSP(P)problem is a tuple(X,C)whereXis the set ofvariablesandCis a distribution over orderedm-tuples ofXreferred to as theconstraints.

An assignment toIis an injective mapO:X→_Zcalled an ordering ofX. For a tuple~c= (v1, . . . ,vm),

O|cdenotes the tuple(O(v1), . . . ,O(vm)). An ordering is said to satisfy the constraintcwhenP(~σ(O|c)) =

valueval(I)of an instance is the maximum value of any ordering. Thus, val(I)def= max

O:X→_Zval(O;I)

def = max

O:X→_ZcE∈_C

P(O|c)

.

To simplify analysis, we extend the predicate toZmand permit assignments which are not necessarily

injective as follows: defineP:Zm→[0,1]by setting P(a1, . . . ,am) =E~σ[P(~σ)] where~σ is drawn

uniformly at random over all permutations inSmsuch thatσi<σjwheneverai<aj. As an example, for

m=3,

P(42,13,13) =1

2P(3,1,2) + 1

2P(3,2,1).

Note that the value of an instance is invariant this extension as there is a complete ordering attaining at least the value of any non-injective assignment.

We define the predicates and problems studied in this work. MAS is exactly OCSP({(1,2)}). The betweenness predicate BTW is the set{(1,3,2),(3,1,2)} and NBTW isS₃\BTW. We define MAX BTW as OCSP(BTW)and MAXNBTW as OCSP(NBTW). Finally, introduce 2t-SO as the subset of

S2t such that the induced ordering on the firsttelements coincides with the ordering of the lasttelements,

2t-SOdef={π∈S₂t|σ(π(1), . . . ,π(t)) =σ(π(t+1), . . . ,π(2t))}.

This predicate has(2t)!/t! satisfying orderings and will be used in proving the inapproximability of the genericm-OCSP with constraints of width at mostm.

2.2 Label cover and inapproximability

The problem LABELCOVER(LC) is a common starting point of strong inapproximability results. An LC instanceL= (U,V,E,L,R,Π)consists of a bipartite graph(U∪V,E)associating with every edge{u,v} a projectionπv,u:R→Lwith the goal of labeling the vertices,λ:U∪V →L∪R, so as to maximize the fraction of projections for whichπv,u(λ(v)) =λ(u). For simplicity, we shall additionally assume that

every projection hasdegreeexactlyd=d(ε). That is, for every LC instance, edgee∈E, and labeli∈L, there are exactlyd choices of j∈Rfor whichπe(j) =i.

Theorem 2.1. For everyε >0, there exists fixed label sets L and R such that it isNP-hard to distinguish

LC instances of value one from instances of value at mostε. Additionally, one can assume that there exists d=d(ε)such that every projection of every instance has degree d.

Proof. Follows from an easy modification [24] of the classical LC construction [2,20].

2.3 Primer on real analysis

We refer to a finite domainΩalong with a distribution µ as a probability space. Given a probability space(Ω,µ), thenthtensor poweris(Ωn,µ⊗n)whereµ⊗n(ω1, . . . ,ωn) =µ(ω1)· · ·µ(ωn). The`pnorm

of f :Ω→ R w. r. t. µ is denoted by ||f||_µ,p and is defined as Ex∼µ[|f(x)|

p

]1/p for real p≥1 and maxx∈supp(µ)f(x)for p=∞. When clear from the context, we omit the distributionµ. The following

Definition 2.2. Let(Ω,µ)be a probability space and f :Ωn→Rbe a function on thenthtensor power.

For a parameterρ∈[0,1], thenoise operatorTρtakes f to Tρf →Rdefined by

Tρf(x)

def

=E[f(y)|x],

where independently for eachi, with probabilityρ,yi=xiand otherwise a uniform random sample.

The noise operator preserves the massE[f]of a function while spreading it in the space. The quantitative bound on this is referred to as hypercontractivity.

Theorem 2.3([25]; [18, Theorem 3.16, 3.17]). Let(Ω,µ)be a probability space in which the minimum nonzero probability of any atom isα<1/2. Then, for every q>2and every f :Ωn→R,

T_ρ(q,α)f

q≤ ||f||2,

where forα <1/2we set

A=1−α α ;

1

q0 =1−

1

q; and ρ(q,α) =

A1/q−A−1/q A1/q0

−A−1/q0

!1/2

.

Forα =1/2, we setρ(q,α) = (q−1)−1/2.

For a fixed probability space, the above theorem says that for everyγ>0, there is aq>2 such that

T₁−γf

q≤ ||f||2. For our application, we need the easy corollary that the reverse direction also holds: for everyγ>0, there exists aq>2 such that hypercontractivity to the`2-norm holds.

Lemma 2.4. Let(Ω,µ)be a probability space in which the minimum nonzero probability of any atom is

α≤1/2. Then, for every f:Ωn→R, small enoughγ>0,

T₁−γf

2+δ ≤ ||f||2

for any

0<δ ≤δ(γ,α) =2log((1−γ)

−2₎₋₁

log(A) with A= 1−α

α >1.

Further,δ(γ,1/2) =γ(2−γ)(1−γ)−2.

Proof. The estimate forδ(γ,1/2)follows immediately from the above theorem assumingγ<1/2. In the case whenα <1/2, solving

ρ2 def= (1−γ)2= (A1/q−A−1/q)(A1−1/q−A1/q−1)−1

forqgives, forγ<1−A−1/2,

δ =q−2= 2 log(A) log ₁₊1+_ρρ22_/A_A

−2≥2

Efron-Stein Decompositions. Our proofs make use of theEfron-Stein decompositionwhich has useful properties akin to Fourier decompositions.

Definition 2.5. Let f:Ω(1)× · · · ×Ω(n)→Randµa measure on∏ Ω(t). TheEfron-Stein decomposition

of f with respect toµis defined as{fS}S⊆[n]where

fS(x)

def =

_∑

T⊆S

(−1)|S\T|_E

f(x0)|x0_T=xT

.

Lemma 2.6(Efron and Stein [10]; Mossel [18]). Assuming{Ω(t)}t are independent, the Efron-Stein

decomposition{fS}Sof f:∏ Ω(t)→Rsatisfies:

• fS(x)depends only onxS,

• for any S,T ⊆[m], andxT ∈∏t∈TΩ(t)such that S\T 6=/0, E[fS(x0)|x0T =xT] =0.

We use the standard notion of influence and noisy influence as in previous work.

Definition 2.7. Let f :Ωn→Rbe a function on a probability space. Theinfluenceof the 1≤ith≤n

coordinate is

Infi(f)

def = E

{Ωj}j6=i

[VarΩi(f)].

Additionally, given a noise parameterγ, thenoisy influenceof theithcoordinate is

Inf(1_i −γ)(f)def= E

{Ωj}j6=i

VarΩi(T1−γf)

.

Influences have simple expressions in terms of Efron-Stein decompositions:

Infi(f) =

∑

Ef_S2 and Inf_i(1−γ)(f) =

_∑

S3i

(1−γ)2|S|Ef_S2.

The following well-known bounds on noisy influences are handy in the analysis.

Lemma 2.8. For everyγ>0, natural numbers i and n such that1≤i≤n, and every f :Ωn→R,

Inf(1_i −γ)(f)≤Var(f) and

_∑

i

Inf(1_i −γ)(f)≤Var(f) γ .

Proof. The former inequality is immediate from the influences in terms of Efron-Stein and Parseval’s Identity for the decomposition:∑SE

f_S2=Var(f). Expanding the LHS of the second inequality,

∑

i

Inf(1_i −γ)(f) =

_∑

k≥1

k(1−γ)2k

_∑

S:|S|=k

E

F_S2

≤Var(f)max

k≥1k(1−γ) 2k

≤Var(f)

_∑

k≥1

(1−γ)2k≤Var(f) 1 (1−γ)2 ≤

Var(f)

Let thetotal (noisy) influenceof a function f:Ωn→Rbe defined as

TotInf(f)def=

_∑

i∈[n]1

Infi(f) and TotInf(1−γ)(f)

def =

_∑

i∈[n]1

Inf(1_i −γ)(f).

Finally, we introduce the notion ofcross influencebetween functions subject to a projection. This notion is implicitly prevalent in modern LC reductions, either for noised functions or for, analytically similar, degree-bounded functions. Given finite setsΩ,L,R; an injective functionπ:L→R; and functions

f :ΩL→R;g:ΩR→R; define

CrInf(1π−γ)(f,g)

def =

_∑

(i,j)∈π

Inf(1_i −γ)(f)Inf(1_j −γ)(g).

Note that our definition differs somewhat from the cross-influence, denoted “XInf,” used by Samorodnit-sky and Trevisan [22].

3

A general hardness result

In this section, we prove a general inapproximability result for OCSPs which, similar to results for classic CSPs, permit us to deduce hardness of approximation based on the existence of certain simple distributions. The proof is via a scheme of reductions from LC to OCSPs. For anm-width predicate

P, we instantiate this scheme with a distributionDoverQt₁×Q₂m−t; for some parameter 1≤t<mand whereQ₁andQ₂are finite integer sets. As an example of a classical distribution,Q₁andQ₂could be the set{−1,1};t=1,m=3; andDcould choose the two first arguments independently, setting the third to be their product. To get appropriate properties for ordering problems, we may however needQ1and

Q₂to be larger ranges than the classical bits, and the sizes of tables in the resulting OCSP instance will similarly depend on these ranges.

When the predicate permits introducing distributions satisfying certain properties, akin to pairwise independence, then the hardness claims in this section follow and, together with the hardness of LC and other properties of the distributions, yield the desired inapproximability results.

The reduction itself is composed of pieces known as dictatorship test which is described in the next section.Section 3.2uses this test to construct the overall reduction and also contains the properties of this reduction. For concrete examples of distributionsD, we refer the reader toSection 4.

3.1 Dictatorship test

The dictatorship test uses the following test distribution (Distribution 1) parametrized byγ, andπand is denoted byT(πγ)(D). The reader should think of the support of the argument distributionDas essentially

being contained in the predicateP.

Recall our convention of extendingPto a predicateP:Zm→[0,1]. For a pair of functions(f,g),

let(f,g)◦(X,Y) denote the tuple (f(~x(1)), . . . ,f(~x(t)_{), g(~y}(t+1)_{), . . . ,g(~y}(m)_{)). Then, theacceptance}

probabilityonT(πγ)(D)for a pair of functions(f,g)where f :Q

L

1→Zandg:QR2→Zis

Accf,g(Tπ(γ)(D))

def

= E

(X,Y)←T(γ) π (D)

Distribution 1:Tπ(γ)(D)

Parameters: • distributionDover

t

z }| {

Q1× · · · ×Q1×

m−t

z }| {

Q2× · · · ×Q2;

• noise parameterγ>0;

• projection mapπ:R→L.

Output: DistributionT(πγ)(D)over

(~x(1), . . . ,~x(t),~y(t+1), . . . ,~y(m))∈ QL₁× · · · ×QL₁× QR₂× · · · ×QR₂.

1. pick a random|L| ×tmatrixXoverQ1by letting each row be a sample fromD≤t,

indepen-dently.

2. pick a random|R| ×(m−t)matrixYdef= (~y(t+1), . . . ,~y(m))overQ₂by letting theithrow be a sample fromD>t conditioned onXπ(i)= (x

(1)

π(i), . . . ,x (t)

π(i))

3. for each entry ofXandY, independently with probabilityγ, replace the entry with a new sample fromQ1andQ2, resp.

4. output(X,Y).

This definition is motivated by the overall reduction described in the next section. The distribution is designed so that functions(f,g)that are dictated by a single coordinate have a high acceptance probability, justifying the name of the test.

Lemma 3.1. Let f :QL₁→Zand g:QR₂ →Zbe defined by g(y) =yi and f(x) =xπ(i)for some i∈R.

Then,

Accf,g(Tπ(γ)(D))≥ E

~x()_∼D[P(x)]−γm.

Proof. The vector(f(~x(1)), . . . ,f(~x(t)),g(~y(t+1)), . . . ,g(~y(m)))simply equals theπ(i)throw ofXfollowed by theithrow ofY. With probability(1−γ)m_≥1−

γmthis is a sample fromDand is hence accepted by

Pwith probability at leastEx∼D[P(x)]−γm.

We prove a partial converse of the above: unless f andghave influential coordinatesiand jsuch that π(j) =i, the distributionDcan be replaced by a product of two distributions with a negligible loss in the acceptance probability. We define this product distribution below and postpone the analysis toSection 5.2 in order to complete the description of the reduction.

Reduction 2:R(_DP_,)_γ

Parameters: distributionDoverQt₁×Qm₂−t and noise parameterγ>0. Input: a Label Cover instanceL= (U,V,E,L,R,Π).

Output: a weighted OCSP(P)instanceI= (X,C)whereX= (U×QL₁)∪(V×QR₂). The distribution of constraints inCis obtained by sampling a random edgee= (u,v)∈Ewith projectionπeand

(X,Y)fromTπ(γe)(D); the constraint is the predicatePapplied to the tuple ((u,~x(1)), . . . ,(u,~x(t)),(v,~y(t+1)), . . . ,(v,~y(m))).

3.2 Reduction from Label Lover

An assignment toIis seen as a collection of functions,{fu}u∈U∪{gv}v∈V, wherefu:QL₁→Z,gv:QR₂→Z

and, e. g., fu(x)is identified with the variable(u,x). The value of an assignment is

E

e=(u,v)∈E;

(X,Y)∈_T(γ) πe(D)

P((fu,gv)◦(X,Y)) = E

e=(u,v)∈E[Accfu,gv(T

(γ)

πe (D))].

Lemma 3.1now implies that ifLis satisfiable, then the value of the output instance is also high. Lemma 3.3. Ifλis a labeling ofLsatisfying a fraction c of its constraints, then the ordering assignment

fu(x) =xλ(u), gv(y) =yλ(v)satisfies at least a fraction

c· E

x∼_D[P(x)]−γm

of the constraints of R(_DP_,)

γ(L). In particular, there is an ordering of R

(P)

D,γ(L)which does not depend onD and attains a value

val(L)· E

x∼_D[P(x)]−γm

.

On the other hand, we also extend the decoupling property of the dictatorship test to the instance output whenval(L)is small. This is the technical core of the paper and is proven inSection 5, more specifically viaLemma 5.9.

Theorem 3.4. Suppose thatDover Qt₁×Qm₂−tsatisfies the following properties: • Dhas uniform marginals;

• for every i>t,Di is independent ofD≤t.

Then, for everyε>0andγ>0there existsεLC>0such that ifval(L)≤εLCthen for every assignment

A={fu}u∈U∪ {gv}v∈V toIit holds that

In particular,

val(R(_DP_,)

γ(L))≤val(R

(P)

D⊥_,γ(L)) +ε.

4

Applications of the general result

In this section, we prove the inapproximability of MAXBTW, MAXNBTW, and MAX2t-SO, using the general hardness result ofSection 3. We also present a gadget reduction from MAXNBTW to MAS.

4.1 Hardness of MAXIMUMBETWEENNESS

Forq∈Z≥1, define the distributionDover{−1,q} ×[q]×[q]by pickingx1∼ {−1,q},y2∼[q], and

y3=

(

y2+1 modq ifx1=q,

y2−1 modq ifx1=−1.

The way to think of this distribution is thatx1will always take on either the smallest or the largest value which in turn dictates whethery3should be smaller or greater thany2in order to satisfy BTW, which will be satisfied when drawn as above except perhaps the unlikelyy2∈ {0,q−1}.

Proposition 4.1. Let(x1,y2,y3)∼D. Then the following holds:

1. Dhas uniform marginals.

2. y2andy3are separately independent ofx.

3. (y2,y3)has the same distribution as(y3,y2).

4. Ex1,y2,y3∼D[BTW(x1,y2,y3)]≥1−1/q.

Letγ>0 be a noise parameter andD⊥ be the decoupled distribution ofDwhich draws the first coordinate independently of the remaining. Consider applyingReduction 2to a given LC instanceLwith test distributionsDandD⊥, obtaining MAXBTW instances

I=RBTW_D,γ (L) and I⊥=RBTW_D⊥_,γ(L).

Lemma 4.2(Completeness). Ifval(L) =1, thenval(I)≥1−1/q−3γ. Proof. This is an immediate corollary ofLemma 3.3andProposition 4.1.

Lemma 4.3(Soundness). For everyε >0,γ>0, q, there is anεLC>0such that ifval(L)≤εLC, then

val(I)≤1/2+ε.

an arbitrary assignment toI⊥. Fix an LC edge{u,v}with projectionπand consider the mean value of constraints produced for this edge by the construction:

E

~x(1)_,y(2)_,y(3)_←T(γ) π (D⊥)

h

BTW

fu(~x(1)),gv(~y(2)),gv(~y(3))

i

. (4.1)

As noted inProposition 4.1,(~y(2),~y(3))has the same distribution as(~y(3),~y(2))when drawn fromD. Consequently, when drawing arguments from the decoupled test distribution, the probability of a specific outcome(x(1),y(2),y(3))equals the probability of(x(1),y(3),y(2)). For strict orderings, possibly having broken ties, at most one of the two can satisfy the predicate BTW. Thus, the expression in(4.1), and in effectval(I⊥), is bounded by 1/2.

Theorem 1.2is now an immediate corollary ofLemmas 4.2and4.3, takingq=d2/εeandγ=ε/6.

4.2 Hardness of MAXIMUMNON-BETWEENNESS

For an implicit parameterq, define a distributionDover[q]3_{by pickingx}

1,y2∼[q]and settingy3= x1+y2 modq.

Proposition 4.4. The distributionDsatisfies the following: 1. Dis pairwise independent with uniform marginals, 2. andEx1,x2,x3∼D[NBTW(x1,y2,y3)]≥1−3/q.

A straightforward application of the general inapproximability witht=1 shows thatx1is decoupled fromy2andy3unlessval(L)is large. Furthermore, pairwise independence implies that the decoupled distribution is simply the uniform distribution over[q]3_{. However, this does not suffice to prove} approxi-mation resistance and in fact the value could be greater than 2/3. To see this, note that if{fu}u∈U,{gv}v∈V

is an ordering of the instance from the reduction, then the first coordinate of every NBTW constraint is a variable of the form fu(·)while the rest aregv(·). Thus, ordering the fu(·)variables in the middle and

randomly orderinggv(·)on both sides satisfies a fraction 3/4 of the constraints.

To remedy this issue and prove approximation resistance, we permute D by swapping the last coordinate with each of the remaining coordinates and overlay the instances obtained by the reduction using these respective distributions. More specifically, for 1≤ j≤3, defineD(j)as the distribution over[q]3obtained by first sampling from Dand then swapping the jth and third coordinate, i. e., the

jthcoordinate is the sum of the other two which are picked independently at random. Similarly, define NBTW(j)as the ordering predicate which is true if the jthargument does not lie between the other two, e. g., NBTW(3)=NBTW.

As in the previous section, take an LC instanceLand consider applyingReduction 2toLwith the distributions{D(j)_}3

j=1, and write

I(j)_=RNBTW(j) D(j)_,γ (L).

Similarly let

I⊥(j)₌

RNBTW(j)

D⊥(j)

be the corresponding decoupled instances.

As the distributionsD(j)_{are over the same domain[q]}3_{, the instancesI}(1)_,I(2)_,I(3)_{are over the same} variables. We define a new instanceIover the same variables as the “sum”

1 3_j

∑

(j)_,

defined by taking all constraints inI(1),I(2),I(3)with multiplicities and normalizing their weights by 1/3. In other words, up to noise, we have an acceptance probability of

E

2

3NBTW(f

u_(x),gv_(y),gv_(xπ₊

qy)) +

1

3NBTW(g

v_(y),gv_(xπ₋

qy),fu(x))

.

Lemma 4.5(Completeness). Ifval(L) =1, thenval(I)≥1−3/q−3γ. Proof. This is an immediate corollary ofLemma 3.3andProposition 4.4.

Lemma 4.6(Soundness). For everyε >0,γ>0, q, there is anεLC>0such that ifval(L)≤εLC, then

val(I)≤2/3+ε.

Proof. Again our goal is to useTheorem 3.4and we start by boundingval(I⊥). To do this, note that the decoupled distributions{D⊥(j)}jare in fact the uniform distributions over[q]3and in particular do

not depend on j. This means that the distributions of variables which NBTW(j)is applied to inI⊥(j) is independent of j, e. g., ifI⊥(1)contains the constraint NBTW(1)(z1,z2,z3)with weightwthenI⊥

(2)

contains the constraint NBTW(2)(z₁,z₂,z₃)with the same weight. In other words,I⊥can be thought of as having constraints of the form

E

j

h

NBTW(j)(z1,z2,z3)

i

.

It is readily verified that

E

j

h

NBTW(j)(a,b,c)i≤2 3

for everya,b,c.

Getting back to the main task of boundingval(I), fix an arbitrary assignment

A={fu:[q]L→Z}u∈U∪ {gv:[q]R→Z}v∈V

4.3 Hardness of MAXIMUMACYCLIC SUBGRAPH

x z y

a

b

Figure 2:The gadget re-ducing NBTW to MAS. The inapproximability of MAS is via a simple gadget reduction from MAX

NBTW. We claim the following properties of the directed graph shownin Fig. 2, defined formally as follows.

Definition 4.7. Define the MAS gadget H from a constraint NBTW(x,y,z)

with auxiliary variables{a,b} as the directed graph H def= (V,A) whereV = {x,y,z,a,b}andAconsists of the walk

b→x→a→z→b→y→a.

Lemma 4.8. Consider an orderingOof x, y, z. Then,

1. ifNBTW(O(x),O(y),O(z)) =1, thenmax_O0val(O0;H) =5/6where the

maxis over all extensionsO0:V→ZofOto V .

2. ifNBTW(O(x),O(y),O(z)) =0, thenmax_O0val(O0;H) =4/6where the

maxis over all extensionsO0ofOto V .

Proof. To find the value of the gadgetH, we individually consider the optimal placement ofaandb

relativex,y,andz. There are three edges in which the respective variables appear: aappears in(x,a), (y,a)and(a,z); whilebappears in(z,b),(b,x), and(b,y).

From this, we gather that two out of the three respective constraints can always be satisfied by placing

aafterx,y,zand similarly placingbbeforex,y,z. We also see that all three constraints involvingacan be satisfied if and only ifzcomes after bothxandy. Similarly, satisfying all three constraints involvingbis possible if and only ifzis ordered before bothxandy. From this, one concludes that if NBTW(x,y,z) =1, i. e., ifzcomes first or last, then we can satisfy five out of the six constraints, whereas ifzis the middle element ofO, we can satisfy only four out of the six constraints.

The proof ofTheorem 1.1is now a routine application of the MAS gadget.

Proof ofTheorem 1.1. Given an instanceIof MAXNBTW, construct a directed graphGby replacing each constraint NBTW(x,y,z)ofIwith a MAS gadgetH, identifyingx,y,zwith the verticesx,y,zofH

and using two new verticesa,bfor each constraint ofI. ByLemma 4.8, it follows that

val(G) =5

6val(I) + 4

6(1−val(I)).

ByTheorem 1.3, it is NP-hard to distinguish betweenval(I)≥1−ε and val(I)≤2/3+ε for every ε>0, implying that it isNP-hard to distinguishval(G)≥5/6−ε/6 fromval(G)≤7/9+ε/6, providing a hardness gap of

7/9 5/6+ε

0

4.4 Hardness of MAXIMUM2t-SAMEORDER

We establish the hardness of MAX2t-SO,Theorem 1.4, via the approximation resistance of the relatively sparse predicate 2t-SO. The proof is similar to the that of MAXBTW inSection 4.1. Letq1<q2be integer parameters and define the base distributionDover[q₁]t×[q₂]t as follows: draw x1, . . . ,xt uniformly

at random from[q1], drawzuniformly at random from[q2], and for 1≤ j≤t setyj=xj+zmodq2.

The distribution of(x1, . . . ,xt,y1, . . . ,yt)definesD. For a permutation~σ∈St, let1σ(·)be the ordering

predicate which is 1 on~σ and 0 on all other inputs.

Proposition 4.9. Dsatisfies the following properties. 1. Dhas uniform marginals.

2. For every i>t,Diis independent ofD≤t.

3. For every~σ∈St,E[1σ(x1, . . . ,xt)] =E[1σ(yt+1, . . . ,ym)] =1/t!.

4. E

x1,...,xt,yt+1,...,ym∼D

[2t-SO(x1, . . . ,xt,yt+1, . . . ,ym)]≥1−

t2

2q1 −q1

q2

.

Proof. The first three properties are immediate from the construction and recalling the extension of predicates to non-unique values. For the last property, note that 2t-SO(x1, . . . ,xt,yt+1, . . . ,ym) =1 if

x1, . . . ,xt are distinct andz<q2−q1. The former event occurs with probability at least 1−t2/(2q1)and the latter independently with probability at least 1−q₁/q₂; a union bound implies the claim.

As in the proof ofTheorem 1.2, letγbe some noise parameter, take an LC instanceL, and let

I=R2_Dt-SO_,γ (L) and I⊥=R2_Dt-SO⊥_,γ(L)

be the respective instances produced byReduction 2using the base distributionDand the decoupled versionD⊥.

The following lemma is an immediate corollary ofLemma 3.3,Proposition 4.9, andItem 4.

Lemma 4.10(Completeness). Ifval(L) =1, thenval(I)≥1− t 2

2q1 −q1

q2 −2tγ.

For the soundness, we have the following.

Lemma 4.11(Soundness). For everyε >0,γ >0,and1≤q1≤q2, there is anεLC>0 such that if

val(L)≤εLC, thenval(I)≤1/t!+ε.

Proof. As in the proof ofLemma 4.3, it suffices to prove thatval(I⊥)≤1/t!. Let{fu:[q1]L→Z}u∈U,

projectionπ. The value of constraints corresponding to{u,v}satisfied by the assignment is

E (X,Y)∈_T(γ)

π (D⊥)

h

2t-SO(fu(~x(1)), . . . ,fu(~x(t)),gv(~y(t+1)), . . . ,gv(~y(2t)))

i

=

_∑

~

σ∈St

E (X,Y)∈T(γ)

π (D⊥)

h

1~σ

fu(~x(1)), . . . ,fu(~x(t))

1~σ

gv(~y(t+1)), . . . ,gv(~y(2t))

i

=

_∑

~

σ∈St E T(γ)

π (D⊥)

h

1~σ

fu(~x(1)), . . . ,fu(~x(t))

i

E T(γ)

π (D⊥)

h

1~σ

gv(~y(t+1)), . . . ,gv(~y(2t))

i

=1/t!,

where the penultimate step uses the independence ofXandYin the decoupled distribution, and the final step usesItem 3ofProposition 4.9.

Theorem 1.4is a corollary ofLemmas 4.10and4.11, taking, e. g.,

q₁=

2t2

ε

, q₂=

2q1

ε

, and γ= ε 8t.

5

Analysis of the reduction

In this section we proveTheorem 3.4which bounds the value of the instance generated by the reduction in terms of the decoupled distribution. Throughout, we fix an LC instanceL, a predicateP, an OCSP instanceIobtained by the procedure R(_DP_,)_γ for a distributionDand noise-parameterγ, and finally an assignmentA={fu}u∈U∪ {gv}v∈V.

The proof involves three major steps. First, we show that the assignment functions, which are

Z-valued, can be approximated by functions on finite domains viabucketing(seeSection 5.1). This

approximation makes the analyzed instance value susceptible to tools developed in the context of finite-domain CSPs [24,6] which are used inSection 5.2to prove the decoupling property of the dictatorship test. Finally, the decoupling is applied to the reduction inSection 5.3and related to the value of the reduced-from LC instance.

5.1 Bucketing

For an integerΓ, we approximate the possibly non-injective function fu:QL₁→Zby partitioning the

range intoΓmultisets. Set q1=|Q1|and partition the cardinality-QL1 multiset range of fu into sets

B₁(fu), . . . ,B(_Γfu) of respective size qL₁/Γsuch that ifx∈B_i(fu) andy∈B(_jfu) for some i< j thenx≤y. Note that this is possible as long as the parameterΓdividesqL₁, which is the case in this treatise. Let

Fu:QL₁ →[Γ]specify the mapping of points to the bucket containing it, and F( a)

u :QL₁ → {0,1} the

indicator of points assigned toB(afu). Partitiongv :Q₂R →Zsimilarly into buckets {B(agv)}obtaining

Gv:QR₂ →[Γ]andG

(a)

We show that a bucketed version approximates well the acceptance probability of the dictatorship test for any edgee= (u,v)fromSection 3,

Accf,g(Tπ(γ)(D))

def

= E

(X,Y)←T(γ) π (D)

[P((f,g)◦(X,Y))].

Fix an edgee= (u,v)and put f= fu,g=gv. As before, we concisely denote by(X,Y)and(f,g)◦(X,Y)

the query tuple and assignment tuple,

(~x(1), . . . ,~x(t),~y(t+1), . . . ,~y(m)) and (f(~x(1)), . . . ,f(~x(t)),g(~y(t+1)), . . . ,g(~y(m))).

Define thebucketed payoff function℘(f,g):[Γ]m→[0,1]with respect to f andgas

℘(f,g)(a1, . . . ,am) = E

~x(i)_←B(f)

ai ;i≤t

~y(j)_←B(g)

a j; j>t

h

P(~x(1), . . . ,~x(t),~y(t+1), . . . ,~y(m))i (5.1)

and thebucketed acceptance probability,

BAccf,g(T

(γ)

π (D)) = E

(X,Y)←_T(γ) π (D)

h

℘(f,g)((F,G)◦(X,Y))

i

. (5.2)

In other words, bucketing corresponds to generating a tuple~a= (F,G)◦(X,Y) and replacing each coordinateaiwith a random value from the bucket whichaifell in. We show that above is close to the

true acceptance probabilityAccf,g(Tπ(γ)(D)).

Theorem 5.1(Bucketing preserves value). For every predicateP, noise parameterγ>0, projection π:R→L, distributionDwith uniform marginals, pair of functions f :Q₁L→Zand g:QR2 →Z, and

bucketing parameterΓ,

Accf,g(T

(γ)

π (D))−BAccf,g(T

(γ)

π (D))

≤m

2

Γ−δ,

for someδ =δ(γ,Q)>0with Q≥max{|Q1|,|Q2|}.

To prove this, we show that for every choice of f and g, including f =g, there is only a few overlapping pairs of buckets, and the probability of hitting any particular pair is small. For eacha∈[Γ],

letR(af)be the smallest interval inZcontainingB(af); and similarlyR( g)

a forg.

Lemma 5.2(Few buckets overlap). For every integerΓthere are at most2Γchoices of pairs(a,b)∈ [Γ]×[Γ]such that R(af)∩R(

g)

b 6=/0.

Proof. Construct the bipartite intersection graphGI = (UI∪VI,EI) where the vertex sets are disjoint

copies of[Γ], and there is an edge (a,b)∈UI×VI iff R

(f)

a ∩R(_bg)6= /0. By construction, the intervals

{R(af)}aare disjoint and similarly{R( g)

b }b. Consequently,GImust be a forest as it does not contain any

Next, we prove a bound on the probability that a fixed pair of themqueries fall in a specific pair of buckets. For a distributionDoverQL₁×Q₂R, defineD(γ)_{as the distribution that samples from}D_{and for}

each of the|L|+|R|coordinates independently with probabilityγreplaces it with a new sample fromD. The distributionD(γ)_{is representative of the projection ofT}(γ)

π (D)to two specific coordinates and we

show that noise prevents the buckets from intersecting with good probability.

Lemma 5.3(Bucket collisions are unlikely). LetDbe a distribution over QL₁×QR₂ whose marginals are uniform in QL₁ and QR₂ andD(γ)_{be as defined above. For every integer}

Γand every pair of functions

F:QL₁→ {0,1}and G:QR₂ → {0,1}such thatE[F(~x)] =E[G(~y)] =1/Γ,

E

(~x,~y)∈D(γ)[F(~x)G(~y)]≤Γ

−(1+δ)

for someδ =δ(γ,Q)>0where Q≥min{|Q1|,|Q2|}.

Proof. Without loss of generality, let |Q₁|=max{|Q₁|,|Q₂|}. Set q=2+δ0 >2 as in Lemma 2.4, 1/q0=1−1/q, and define

H(~x)def= E

~y|~x

T1−γG(~y)

.

Then,

E

(~x,~y)∈_D(γ)[F(~x)G(~y)] =E_~x

T1−γF(~x)H(~x)

≤ T_1−γF

q||H||q0

≤ ||F||₂||H||_q0 ≤ ||F||₂

T₁−γG

q0≤ ||F||₂||G||_q0

=Γ−(1/2+1/q 0₎

=Γ−(1+δ 0

/2(2+δ0))_,

using the hypercontractivity fromLemma 2.4and Jensen’s inequality.

Note that the above lemma applies to queries to the same function as well, settingF=G, etc. To complete the proof ofTheorem 5.1, we apply the above lemma to every distinct pair of themqueries made inTπ(γ)(D)and each of the at most 2Γpairs of overlapping buckets.

Proof ofTheorem 5.1. Note that the bucketed payoff℘(f,g)((F,G)◦(X,Y))is equal to the true payoff

P((f,g)◦(X,Y))except possibly when at least two elements in(F,G)◦(X,Y)fall in an overlapping pair of buckets. Fix a pair of inputs, sayx(i)andy(j)– the proofs are identical when choosing two inputs fromXor two fromY. Leta=F(x(i))andb=G(y(j)). ByLemma 5.2there are at most 2Γpossible

values(a,b)such that the buckets indexed byaandbare overlapping. FromLemma 5.3, the probability thatF(~x(i)) =aandG(~y(j)) =bis at mostΓ−1−δ. By a union bound, the two outputsF(~x(i)),G(~y(j))

consequently fall in overlapping buckets with probability at most 2Γ−δ. As there are at most m₂≤m2/2

5.2 Soundness of the dictatorship test

We now reap the benefits of bucketing and prove the decoupling property of the dictatorship test alluded to inSection 3.

Theorem 5.4. For every predicatePand distributionDsatisfying the conditions ofTheorem 3.4, and any noise rateγ>0, projectionπ:R→L, and bucketing parameterΓ, the following holds. For any functions f :QL₁→Z, g:QR2→Zwith bucketing functions F :QL1→[Γ], G:QR2 →[Γ],

BAccf,g(T

(γ)

π (D))−BAccf,g(T

(γ)

π (D

⊥

))

≤γ

−1/2_m1/2₄m

Γm

_∑

a,b∈[Γ]

CrInf(1π−γ)

F(a),G(b)

1/2

.

Recall that the decoupled versionD⊥ of a base distributionDis obtained by combining two inde-pendent samples ofD: one for the firsttcoordinates and one for the remaining. A similar claim, as above, for the true acceptance probabilities of the dictatorship test is now a simple corollary of the above theorem andTheorem 5.1. This will be used later in extending the decoupling property to our general inapproximability reduction.

Lemma 5.5. For every predicatePand distributionDsatisfying the conditions ofTheorem 3.4, and any noise rateγ>0, projectionπ:R→L, and bucketing parameterΓ, the following holds. For any functions

f :QL₁→_Z, g:QR₂→_Zwith bucketing functions F:QL₁ →[Γ], G:QR₂ →[Γ],

Accf,g(T

(γ)

π (D))−Accf,g(T

(γ)

π (D

⊥

))

≤γ−1/2m1/24mΓm

_∑

a,b∈[Γ]

CrInf(1π−γ)

F(a),G(b)

1/2

+2Γ−δm2. (5.3)

The proof of the theorem is via an invariance-style theorem and uses techniques found in the works of Mossel [18], Samorodnitsky and Trevisan [22], and Wenner [24]. Frankly, it mostly involves introducing new notation to apply existing machinery. The notion of lifted functions may be the most alien: a large-side tableg:[q2]R→Rmay equivalently be seen as the functiong0π:Ω0L→RwhereΩ0 = [q2]d contains the values of alldcoordinates inRprojecting to the same coordinate inL.g0πis called the lifted analogue ofgwith respect to the projectionπ.

The first lemma proved in this section essentially says that if a product of functions is influential, then at least one of the functions is also influential. The second says that if the lifted analogue of a functiong

is influential for some coordinatei∈L, thengis influential for a coordinate projecting toi. The lemma will be used to—after massaging the expression—decouple the small-side table from the lifted analogue of the large-side table as a function of their cross influence.

Lemma 5.6(Mossel [18, Lemma 6.5]). Let f1, . . . ,ft :Ωn→[0,1]be arbitrary functions. Then for any

j∈[n],Infj(∏tr=1fr)≤t∑tr=1Infj(fr).

Lemma 5.7. Consider a function g:ΩR→R, a mapπ:R→L, and an arbitrary bijectionς :R↔ ((`)×

|π−1_(`)|

)`∈L such that for all(i,j),∃tς(j) = (i,t) iffπ(j) =i. IntroduceΩ0<sub>`</sub>

def =Ω|π

−1<sub>(`)|</sub> and define g0:∏LΩ0`→Ras g0(~z) =g(~y)where yj=zς(j). Then for every i∈L,

Infi g0

≤

_∑

j∈π−1(i)

Proof. Using definitions and the law of total conditional variance,

Infi g0

= E

Ω0−i

Var

Ω0i

g0

= E

{Ω`}`:6∃_jς(i,j)=`

"

Var

{Ω`}`:∃_jς(i,j)=`

g

#

≤

_∑

`:∃jς(i,j)=`

E Ω−` Var Ω` g =

_∑

`:∃jς(i,j)=`

Inf`(g).

Theorem 5.8(Wenner [24, Theorem 3.21]). Consider functions{f(r)∈L∞₍

Ωnr)}r∈[m]₁ on a probability

spaceP= (∏mi=1Ωi,P)⊗n, a set M([m]1, and a collectionCof minimal sets1C⊆[m]1,C*M such that

the spaces{Ωi}i∈Care dependent. Then,

E " m

∏

f(r)

#

−

_∏

r∈/M

Ehf(r)

i

E

"

∏

r∈M

f(r)

#

≤4mmax

C∈_C s

min

r0_∈CTotInf f

(r0₎

∑

∏

Inf` f(r)

∏

r∈/C

f

(r) ∞ .

Proof ofTheorem 5.4

Proof. We massage the expressionBAccf,g(Tπ(γ)(D))to a form suitable for applyingTheorem 5.8. Recall

thatF(a)denotes the indicator of “F(x) =a” and similarlyG(a)of “G(y) =a.” In terms of these indicators, BAccf,g(T

(γ)

π (D))equals

∑

℘(~a,~b) E T(γ)

π (D)

"

t

∏

r=1

F(ar)(x(r))

m

∏

r=t+1

G(br−t)_(y(r)₎

#

.

Consequently,BAccf,g(D)−BAccf,g(D⊥)

may be bounded from above by

∑

℘(~a,~b)

E T(γ)

π (D)

"

t

∏

r=1

F(ar)(x(r))

m

∏

r=t+1

G(br−t)_(y(r)₎

#

− E

T(γ) π (D⊥)

"

t

∏

r=1

F(ar)(x(r))

m

∏

r=t+1

G(br−t)_(y(r)₎

# . (5.4)

We note that 0≤℘≤1 and proceed to bound the difference in the summation for fixed(a,b). To this end, we make a slight change of notation as discussed previously. The new notation may seem cumbersome; the high-level picture is that we group the first set of functions into a single function, receiving a single argument over a larger domain, and redefine the latter functions to take arguments indexed byLinstead ofR. Also recall that we intend to reduce from LC instances with projection degrees exactlydfor somed=d(εLC).

Definem0=m−t+1,Ω1= [q1]t,Ω2=· · ·=Ωm0= [q₂]d. Let_ς be a bijectionL×[d]↔Rsuch that

π(ς(i,j0)) =i; i. e., for eachi∈L, we group together thedcoordinates inRmapping toi. Introduce the distribution

ΩL₁× · · · ×ΩL_m0 3(~w,z(2), . . . ,z(m 0₎

)∼R(µ)

1_{Minimal sets of dependent outcome spaces in the sense that ifC∈C, then the outcomes spaces of every strict subset ofC}

which samples(x(1), . . . ,x(t),y(t+1), . . . ,y(m))fromTπ(γ)(D), setting

wi,r=x( r)

i and z

(r)

i = (y

(r)

ς−1(i,j0₎)

d−1

j0₌₀.

Let

W(~w)def=

t

∏

r=1 T1−γF

(ar)_(x(r)₎

wherex(_ir)=wi,rand similarly, for 2≤r≤m0, call the lifted functionH(r):ΩLr →Rdefined as

H(r)(z) = T1−γG

(br−1)_(y)

wherey_ς(i,j0₎=z_i,j0. In the new notation, the difference within the summation in (5.4) is

E R(D)

"

W(~w)

m0

∏

r=2

H(r)(z(r))

#

− E R(D⊥₎

"

W(~w)

m0

∏

r=2

H(r)(z(r))

# . (5.5)

We note thatRis a product distributionR=µ⊗Lfor someµand for any 2≤r≤m0,Ωris independent

ofΩ1due to the premise thatDrindependent ofD≤t. ChoosingM={2, . . . ,m0}, minimal indicesCof

dependent sets inµ not contained inMcontains 1 and at least two elements fromM, i. e.,C∈Cimplies 1,e,e0∈Cfor somee6=e0∈ {2, . . . ,m0}.

ApplyingTheorem 5.8and choosingr06=1 bounds the difference (5.5) by

4m

s

max

C∈C1min6=e∈CTotInf(H

(e)₎

∑

i

Infi(W)

∏

Infi H(t)

∏

r∈/C

H

(r) ∞ . (5.6)

As we assumed the codomain of the studied functions{G(r)}rto be[0,1]the same holds for{H(r)}r

and consequently the influences and infinity norms in (5.6) are upper-bounded by one on account of Lemma 2.8, yielding at most

(5.6)≤4m max

e6=1 TotInf(H

(e)_)·max

e6=1

∑

H(e)

!1/2

. (5.7)

We recall thatW=∏tr=1T1−γF(ar)and hence byLemma 5.6,

Infi(W)≤t t

∑

r=1 Infi

T1−γF

(ar)_=t

∑

F(ar)

.

Similarly,Lemma 5.7implies that

Infi

H(e)

≤

_∑

j∈π−1(i) Infj

T1−γG(

be−1)

=

_∑

j∈π−1(i)

Inf(1j −γ)

G(be−1)

Returning to Equation (5.7), we have the bound

(5.7)≤4m tmax

e>t TotInf

(1−γ)_(G(be)_)·max

e>t

∑

t

∑

r=1

Inf(1_i −γ)

F(ar)

∑

j∈π−1(i)

Inf(1_j −γ)

G(be)

!1/2

. (5.8)

Usingt≤m; maxe· · · ≤∑e· · · for non-negative expressions; bounding the total noisy influence byγ−1

fromLemma 2.8; and identifying the inner sum as a cross influence, we establish the desired bound on the considered difference as

(5.5)≤(5.8)≤4m tγ−1

_∑

r,e

CrInf(1π−γ)

F(ar),G(be)

!1/2

≤γ−1/2m1/24m

_∑

r,r0

CrInf(1π−γ)

F(ar),G(br0)

1/2

.

Finally, as we noted before, since 0≤℘≤1, and since there are at mostΓmterms in the summation,

(5.4) is at most

γ−1/2m1/24mΓm

_∑

a,b∈[Γ]

CrInf(1π−γ)

F(a),G(b)

1/2

.

5.3 Soundness of the reduction

With the soundness for the dictatorship test in place, proving the soundness of the reduction(Theorem 3.4) is a relatively standard task of constructing noisy-influence decoding strategies.

The proof follows immediately from the more general estimate given in the following lemma by taking

Γ=

l

(4m2/ε)1/δ

m

and then εLC=

ε γ3/2

m1/2₄m

Γm+1

!2

.

Lemma 5.9. Given an LC instanceL= (U,V,E,L,R,Π)and a collection of functions, fu:QL₁→Zfor

u∈U ; gv:QR₂→Zfor v∈V , andΓ,γ,δ as in this section,

E

u,v∼E

h

Accfu,gv(T (γ)

π (D))−Accfu,gv(T (γ)

π (D

⊥₎₎ i

≤γ−1.5m1/24mΓm+1val(L)1/2+2Γ−δm2.

Proof. For a function f :QL₁ → _Zdefine a distribution Ψ(f) over L as follows. First picka∼[Γ]

uniformly, then pick`∈Lwith probabilityγ·Inf<sub>`</sub>(1−γ)(Fv(a))and otherwise an arbitrary label. Note that

byLemma 2.8,

∑

Inf_`(1−γ)(Fu(a))≤1/γ

and so picking`∈Lwith the given probabilities is possible. DefineΨ(g)overRforg:QR₂ →_Zsimilarly. Now define a labeling ofLby, for eachu∈U (andv∈V), sampling a label fromΨ(fu)(Ψ(gv), resp.),

For an edgee= (u,v)∈E, the probability thateis satisfied by the labeling equalsP(πe(Ψ(fu)) =

Ψ(gv)), which can be lower-bounded by

∑

i,j:πe(j)=i

γ2 E

a,b∈[Γ]

h

Inf(1i −γ)

Fu(a)

Inf(1j−γ)

G(vb)

i

= (γ/Γ)2

_∑

a,b

CrInf(1πe−γ)

Fu(a),G( b)

v

.

Taking the expectation over all edges ofL, we get that the fraction of satisfied constraints is

(γ/Γ)2 E

e=(u,v)

"

∑

a,b

CrInf(1πe−γ)

Fu(a),G(vb)

#

≤val(L),

and by concavity of the√·function, this implies that

E

e=(u,v)

"

∑

a,b

CrInf(1πe−γ)

Fu(a),G

(b)

v

1/2 #

≤Γγ−1val(L)1/2.

Plugging this bound on the total cross influence into the soundness for the test,Lemma 5.5, we obtain

E

e=(u,v)

h

Accfu,gv(T (γ)

πe (D))−Accfu,gv(T (γ) πe (D

⊥₎₎ i

≤γ−1.5m1/24mΓm+1val(L)1/2+2Γ−δm2,

as desired.

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AUTHORS

Per Austrin associate professor

KTH – Royal Institute of Technology, Stockholm, Sweden austrin kth se

http://www.csc.kth.se/~austrin

Rajsekar Manokaran postdoc

KTH – Royal Institute of Technology, Stockholm, Sweden rajsekar csc kth se

Cenny Wenner Ph. D. student

KTH – Royal Institute of Technology, Stockholm, Sweden cenny cwenner net

http://www.cwenner.net

ABOUT THE AUTHORS

PERAUSTRINgraduated from KTH – Royal Institute of Technology in Sweden, 2008. His advisor was Johan Håstad. Following this, he did post-docs at Institut Mittag-Leffler, New York University, the University of Toronto, and Aalto University. In his spare time he enjoys writing tepid biographic blurbs.

RAJSEKARMANOKARANreceived his Ph. D. from Princeton University under the supervi-sion of Sanjeev Arora. He is currently a postdoctoral researcher with Johan Håstad at KTH in Stockholm.