Tight Hardness Results for Some Approximation Problems [Håstad]

Tight Hardness Results for Some Approximation Problems [Håstad]
Adi Akavia Danna Moshkovits S. Safra 33

“Road-Map” of the Presentation
() = ()  Parallel repetition lemma LLC-Lemma: (L) = ½+/2  (par[,k]) > 42 expander Long code L  par[, k]  Gap-3-SAT-7  Gap-3-SAT  ©Safra,Akavia,Moshkovitz

©Safra,Akavia,Moshkovitz
Maximum Satisfaction Def: Max-SAT Instance: A set of variables Y = { Y1, …, Ym } A set of Boolean-functions (local-tests) over Y  = { 1, …, l } Maximization: We define () = maximum, over all assignments to Y, of the fraction of i satisfied Structure: Various versions of SAT would impose structure properties on Y, Y’s range and  ©Safra,Akavia,Moshkovitz

Max-E3-Lin-2 Def: Max-E3-Lin-2 Instance: a system of linear equations L = { E1, …, En } over Z2 each equation of exactly 3 variables (whose sum is required to equal either 0 or 1) Problem: Compute (L) ©Safra,Akavia,Moshkovitz

Example x1+x2+x3=1 (mod 2) x4+x5+x6=1 x7+x8+x9=1 x1+x4+x7=0 x2+x5+x8=0 x3+x6+x9=0 Assigning x1-6=1, x7-9=0 satisfy all but the third equation. No assignment can satisfy all equation, as the sum of all leftwing of equations equals zero (every variable appears twice) while the rightwing sums to 1. Therefore, (L)=5/6. ©Safra,Akavia,Moshkovitz

Main Theorem Thm: gap-Max-E3-Lin-2(1-, ½+) is NP-hard. That is, for every constant 0<<¼ it is NP-hard to distinguish between the case where 1- of the equations are satisfiable and the case where ½+ are. [ It is therefore NP-Hard to approximate Max-E3-Lin-2 to within factor 2- for any constant 0<<¼] ©Safra,Akavia,Moshkovitz

This bound is tight A random assignment satisfies half of the equations. Deciding whether a set of linear equations have a common solution is in P (Gaussian elimination). ©Safra,Akavia,Moshkovitz

Gap-3-SAT Proof Outline expander Gap-3-SAT-7 Parallel repetition lemma () = ()  par[, k] Def: 3SAT is SAT where every i is a disjunction of 3 literals. Def: gap-3SAT-7 is gap-3SAT with the additional restriction, that every variable appears in exactly 7 local-tests Theorem: gap-3SAT-7 is NP-hard The proof proceeds with a reduction from gap-3SAT-7(1, 1-) for some constant >0, known to be NP-hard Given such an instance, the proof shows a poly-time construction, of an instance of Max-E3-Lin-2 with the claimed (1-, ½+) gap. Long code L  LLC-Lemma: (L) = ½+/2  (par[,k]) > 42  proof ©Safra,Akavia,Moshkovitz

Distributional Assignments
Gap-3-SAT expander Distributional Assignments () = ()  Parallel repetition lemma Gap-3-SAT-7 par[, k] Long code L  Consider a SAT instance  over variables X of range R. Let (R) be the set of all distributions over R Def: A distributional-assignment to  is A: X  (R) Denote by () the maximum over distributional-assignments A of the average probability for    to be satisfied, if variables` values are chosen according to A Clearly ()  (). Moreover Prop: ()  () LLC-Lemma: (L) = ½+/2  (par[,k]) > 42  ©Safra,Akavia,Moshkovitz

Notations Def: For a 3SAT formula  over Boolean variables Y, Let Yk be the set of all k-sequences of ’s variables Let k be the set of all k-sequences of ’s clauses Def: For any VYk and Ck, let SV be the set of all assignments to V SC be the set of all assignments to C Def: For any set of k variables VYk, and a set of k clauses Ck, denote V C  V is a choice of one variable of each clause in C. ©Safra,Akavia,Moshkovitz

Restriction and Extension
Def: For any VYk and Ck s.t V C, The natural restriction of an aSC to SV is denoted a|V The elevation of a subset FP[SV] to SC is the subset F*P[SC] of all C assignments whose restriction to V is in F F* = { a | a|V  F } ©Safra,Akavia,Moshkovitz

Gap-3-SAT Parallel SAT expander Parallel repetition lemma Gap-3-SAT-7 par[, k] () = ()  Long code Def: For a 3SAT formula  over Boolean variables Y, denote by par[, k] the following SAT instance: par[, k] has two types of variables: x[V] for every set VYk, where x[V]‘s range is the set SV of all assignments to V x[C] for every set Ck, where x[C]‘s range is the set SC of all satisfying assignments to all clauses in C par[, k] has one local-test, [C,V], for every V C: [C,V] accepts if x[C]|V = x[V] (namely, if the assignments to x[C] and x[V] are consistent) L  LLC-Lemma: (L) = ½+/2  (par[,k]) > 42  |SV|=2k |SC|=7k ©Safra,Akavia,Moshkovitz

Gap Increases with k Note that if () = 1 then (par[, k]) = 1
Parallel repetition lemma Gap-3-SAT Gap Increases with k expander Gap-3-SAT-7 () = ()  par[, k] Note that if () = 1 then (par[, k]) = 1 On the other hand, if  is not satisfiable: Lemma: (par[, k])  ()c·k for some c>0 Proof: first note that 1-(par[, 1])  (1-())/3 now, to prove the lemma, apply the Parallel-Repetition lemma [Raz] to par[, 1] Long code L  LLC-Lemma: (L) = ½+/2  (par[,k]) > 42  In any assignment to ‘s variables, any unsatisfied clause in  ”induces“ at least 1 (out of corresponding 3) unsatisfied  par[, 1] ©Safra,Akavia,Moshkovitz

Usually a code is regarded as a sequence, but here we ignore the order between the functions. Long-Code over Range R Any binary code C, of elements of a domain R, can be represented by a set C of Boolean-functions (B.f.) f: R{0,1}. Def: A C-encoding is a binary assignment A: C {0,1} Def: A C-encoding A: C {0,1} is a C-legal-encoding of an element aR, if f  C A(f) = f(a). The most extensive binary-code of aR lists f(a) for all B.f. f over R ©Safra,Akavia,Moshkovitz

Gap-3-SAT Long-Code over Range R expander Parallel repetition lemma Gap-3-SAT-7 () = ()  par[, k] Long code BP[R]  the set of all subsets of R of size ≤½|R| Def: an R-long-code has 1 bit for each F  P[R] namely, any boolean function: P[R]  {0, 1} Def: a legal-long-code-word of an element aR, is a long-code ERa: P[R]  {0, 1} that assigns aF to every subset F  P[R] |BP[R]| = 2|R|-1-1 L  LLC-Lemma: (L) = ½+/2  (par[,k]) > 42 Our long-code: in our context there’re two types of domains “R”: SC for a set C of k clauses of , SV for a set V of k variables of   The most extensive binary-code ©Safra,Akavia,Moshkovitz

Encoding an Element In R By Its Long-Code
Given eR, list f(e) for all Boolean functions f over R correction: 2n-1-1 example for e=r1 1 1 f0 f1 f2 f2n-1-1 f2n-1 r1 r2 ... rn • 1 r1 r2 ... rn • 1 r1 r2 ... rn • 1 r1 r2 ... rn • 1 Note: 1) Each entry determines uniquely the value of the entry corresponding to the complement function. 2) The value of the entry corresponding to the constant function 0 (1) is always 0 (1). ©Safra,Akavia,Moshkovitz

Linearity of a Legal-Encoding
An assignment A : BP[R]  {0,1}, if legal, is a linear-function, i.e.,  F, G  BP[R]: F + G  ‘FG’ (mod 2) where ‘FG’  BP[R] is the symmetric difference of F and G Unfortunately, the bit-wise xor of two legal encodings of a,b  R, is linear as well perhaps ‘FGR’  BP[R] FG  R ‘FGR’ ©Safra,Akavia,Moshkovitz

The Variables of L Consider par[,k] for large
Gap-3-SAT The Variables of L expander Parallel repetition lemma Gap-3-SAT-7 The constructed linear-equation-system () = ()  par[, k] Consider par[,k] for large constant k (to be fixed later) L has 2 types of variables: a variable z[V,F] for every variable x[V] of par[,k] and a subset F  BP[SV] a variable z[C,F] for every variable x[C] of par[,k] and a subset F  BP[SC] Long code L  LLC-Lemma: (L) = ½+/2  (par[,k]) > 42  Thus every L-variable corresponds to a subset of either: SC (the set of all assignments to k variables of ), or SV (the set of all satisfying assignments to k clauses of ) ©Safra,Akavia,Moshkovitz

The Distribution  Def: denote by  the distribution over all subset of SC, which assigns probability to a subset H as follows: Independently, for each a  SC, let aH with probability 1- aH with probability  One should think of  as a multiset of subsets in which every subset H appears with the appropriate probability ©Safra,Akavia,Moshkovitz

Linear equation L‘s linear-equations are the union, over all [C,V]  par[,k], of the following: F  BP[SV], G  BP[SC] and H   F G = ‘F*GH’ i.e.,  [C,V], F, G, H there’s a test z[V,F] + z[C, G] = z[C,’F*  G  H’] (mod 2) ‘F*GH’ is the symmetric difference of the extension of F to SC, G and H ©Safra,Akavia,Moshkovitz

Prop: if () = 1 then (L) = 1-
Gap-3-SAT expander Parallel repetition lemma Gap-3-SAT-7 () = ()  par[, k] Long code Prop: if () = 1 then (L) = 1- Proof: Let A be a satisfying assignment to par[,k]. Assign all variables of L according to the legal encoding of A’s values. A linear equation of L, corresponding to C,V,F,G,H, would be unsatisfied exactly if h(x[C])=1, which occurs with probability  over the choice of H. LLC-Lemma: (L) = ½+/2  (par[,k]) > 42 L  LLC-Lemma: (L) = ½+/2  (par[,k]) > 42  Note: independent of k! (Later we use that fact to define k large enough for our needs).  = 2(L) -1 ©Safra,Akavia,Moshkovitz

Hardness of approximation of Max-E3-Lin-2
Main Theorem: For any constant >0: gap-Max-E3-Lin-2(1-,½+) is NP-hard. Proof: Let  be a gap-3SAT-7(1, 1-) By proposition () = 1  (L)  1-  ©Safra,Akavia,Moshkovitz

of the parallel repetition lemma
Lemma  Main Theorem Prop: Let  be a constant >0 s.t.: (1-)/(½+/2)  2- Let k be large enough s.t.: 43 > ()c·k Then () < 1  (L)  ½+/2  ½+  Proof: Assume, by way of contradiction, that (L)  ½+/2 then: 43 > ()c·k  (par[, k]) > 42, which implies that  > . Contradiction! of the parallel repetition lemma ©Safra,Akavia,Moshkovitz

Road-Map for the Proof of LLC-Lemma
Multiplicative representation  General Fourier Analysis facts  Representation by Fourier Basis  Claim 2:  E[C,V] [  Sc FV|V• (FC)2 • (1-2)|| ] =  Claim 1:  success probability of  on [C,V]par[,k] is  Sc (FV|V)2 • (FC)2 • ||-1 Claim 3:  The expected success of the distributional assignment  on [C,V]par[,k] is at least 4 2 (par[,k]) > 42  ©Safra,Akavia,Moshkovitz

Inner Product Spaces Def: W is an Inner Product Space
General Fourier Analysis facts Inner Product Spaces Multiplicative representation Representation by Fourier Basis Def: W is an Inner Product Space if W is a vector-space (over ) with an inner product operator: <.,.> : W  W   Such that <a+b, c> = <a+c> + <b+c> <a, b> = <a, b> a0 <a, a> > 0 Claim 2 Claim 1 Claim 3:The expected success of the distributional assignment  on [C,V]par[,k] is at least 4 2 (par[,k]) > 42  ©Safra,Akavia,Moshkovitz

Orthonormal Basis Def: O is an Orthonormal System if a,bO a  b, and aO ||a||=1 i.e.: 1 if a=b <a,b> = 0 otherwise Def: an orthonormal system {ui}i=1...n is an Orthonormal Basis of an inner-product space W, if aW, a  i=1...n<a,ui>ui ©Safra,Akavia,Moshkovitz

Parseval’s Formula Thm: Let W be an inner-product space, and {ui}i=1...n be an orthonormal basis for W, then a W ||a||2 = i=1...n<a,ui>2 ©Safra,Akavia,Moshkovitz

Revised Representation
Multiplicative representation General Fourier Analysis facts Revised Representation Representation by Fourier Basis Multiplicative Representation: 1 (True)  -1 0 (False)  1 f + g  f  g L: z[C,*], z[V,*]  {-1, 1} z[V,f] • z[C, g] • z[C,’f•g•h’] = 1 Claim 2 Claim 1 Claim 3:The expected success of the distributional assignment  on [C,V]par[,k] is at least 4 2 (par[,k]) > 42  ©Safra,Akavia,Moshkovitz

Long Code - Revised Representation
Note, that in those revised notations, Boolean functions evaluate to either 1 or –1 Def: The long-code of R has one bit for every B.f. in BP[R] namely, BP[R]  {1, -1} The legal-encoding of an element aR, is the mapping: ERa: BP[R]  {-1,1} , where ERa(f) = f(a). i.e. ERa assigns f(a) to every B.f. fBP[R]. ©Safra,Akavia,Moshkovitz

Long-Code as an inner product space
Def: Let R = { A : BP[R]  {-1,1} } R is an inner-product space:  A , B  R ©Safra,Akavia,Moshkovitz

Basis for Long-Code go to claim3 Def: For every   R, let Claim: For a domain R, the set: {  = af(a) :   R } is an orthogonal basis of R Corollary:  A  R A =   R < A ,  > (see Basis def)  <A, >2 = 1/|BP[R]|·fBP[R] (A(f))2 = 1 (by Parseval’s formula and the fact the A(f){-1,1}) Note, now we can use <A, >2 as a probability measure! ©Safra,Akavia,Moshkovitz

Representation with Fourier Basis
General Fourier Analysis facts Representation with Fourier Basis Multiplicative representation Representation by Fourier Basis The representation of A  R as: A =   R < A ,  >· is called: “the Fourier representation of A” The coefficients < A , > are called the Fourier-coefficients of A Claim 2 Claim 1 Claim 3:The expected success of the distributional assignment  on [C,V]par[,k] is at least 4 2 (par[,k]) > 42  ©Safra,Akavia,Moshkovitz

An Assignment  to L For any set C of k clauses of  The set z[C,*] of variables of L represent the long-code of x[C] Let FC be the Fourier-Coefficient <|z[C,*],> For any set V of k variables of  The set z[V,*] of variables of L represent the long-code of x[V] Let FV be the Fourier-Coefficient <|z[V,*],> ©Safra,Akavia,Moshkovitz

The Distributional Assignment.
Def: Let  be a distributional-assignment to par[,k] as follows: For any variable x[C] Choose a set SC with probability (FC)2, Uniformly choose a random assignment a. For any variable x[V] Choose a set SV with probability (FV)2, Uniformly choose a random assignment b. ©Safra,Akavia,Moshkovitz

Longcode and Fourier Coeficients
go to claim2 Auxiliary Lemmas: 1. For any f,gBP[R] and   R, (f·g) = (f)·(g). 2. For any fBP[R] and ,  R, (f)·(f) = (f), where  is the symmetric difference between  and . 3. For any random f (uniformly chosen) and , E[ (f) ]=0 and E[ (f) ]=1. =xf(x) apply multiplication’s commutative & associative properties (f)·(f)=xf(x)·xf(x)= xf(x)2·x(f)=1·x(f) x, f(x) is 1 or -1 with probability ½ ©Safra,Akavia,Moshkovitz

Home Assignment Given an assignment to a Longcode A:BP[R]  {0, 1}, show that for any (constant)  > 0, there is a constant h(), which depends on , however does not depend on R such that: | {e  R | (Ee, A) > ½ +  } |  h() where (A1, A2) is the fraction of bits A1 and A2 differ on. ©Safra,Akavia,Moshkovitz

What’s Ahead: We would show that ‘s expected success on [C,V]par[,k] is > 42 in two steps: First we show (claim 1) that ‘s success probability, for any [C, V]  par[,k] is  Sc (FV|V)2 • (FC)2 • ||-1 Then show (claim 3) that value to be  42 ©Safra,Akavia,Moshkovitz

Claim 1 Claim 1: The success probability of  on [C,V]par[,k] is
General Fourier Analysis facts Claim 1 Multiplicative representation go to claim3 Representation by Fourier Basis Claim 1: The success probability of  on [C,V]par[,k] is  Sc (FV|V)2 • (FC)2 • ||-1 Proof: That success probability is at least Sv,Sc (FV)2 • (FC)2 • Prb[b|V ] and if =|V there is at least one b s.t. b|V   So, ‘s success probability is at least ||-1 times the case in which the chosen  and  satisfy: |V = , i.e. at least Sc (FV|V)2 • (FC)2 • ||-1 Claim 1 Claim 2 Claim 3:The expected success of the distributional assignment  on [C,V]par[,k] is at least 4 2 (par[,k]) > 42  ©Safra,Akavia,Moshkovitz

Auxiliary Lemmas go to claim3 ...Some More Auxiliary Lemmas: For any positive constants a and s: a-se-sa e-x  1-x E[x2]  E[x]2 (proof: 0  E[ (x-E[x])2 ] = E[ x2-2E[x]x+E[x]2 ] = E[x2] - 2E[x]E[x] + E[x]2 = E[x2] - E[x]2 ) ©Safra,Akavia,Moshkovitz

Lemma’s Proof - Claim 2 (1)
go to claim3 General Fourier Analysis facts Claim 2: E[C,V][ Sc FV|V•(FC)2•(1-2)||] =  Proof: The test accepts iff z[V,f]•z[C, g]•z[C,’f•g•h’] = 1 By our assumption, this happens with probability /2+½. Now, according to the definition of the expectation: E[C,V], f, g, h[z[V,f]•z[C, g]•z[C,’f•g•h’]] = 1•(½+/2) + (-1)•(1 -(½+/2)) =  Multiplicative representation Representation by Fourier Basis Claim 2 Claim 1 Claim 3:The expected success of the distributional assignment  on [C,V]par[,k] is at least 4 2 (par[,k]) > 42  ©Safra,Akavia,Moshkovitz

Lemma’s Proof - Claim 2 (2)
We next show that Ef,g,h[z[V,f]•z[C, g]•z[C,’f•g•h’]] = ScFV|V• (FC)2•(1-2)|| Hence, E[C,V], f, g, h[z[V,f]•z[C, g]•z[C,’f•g•h’] ] = E[C,V] [ Ef,g,h[z[V,f]•z[C, g]•z[C,’f•g•h’] ] ] = E[C,V] [ ScFV|V • (FC)2 • (1-2)|| ]  E[C,V] [ Sc FV|V • (FC)2 • (1-2)|| ] =  ©Safra,Akavia,Moshkovitz

Lemma’s Proof - Proposition
By Fourier transform auxiliary lemma 2 auxiliary lemma 1 auxiliary lemma 3 Expectation Linearity

Lemma’s Proof - Claim 3 Claim 3: The expected success of the distributional assignment  on [C,V]par[,k] is at least 42 Proof: Claim 1 gives us the initial lower bound for the expected success: ©Safra,Akavia,Moshkovitz

Lemma’s Proof - Claim 3 As we’ve already seen, (FC)2=1. Hence, our lower-bound takes the form of Or alternatively, Which allows us to use the known inequality E[x2]E[x]2 and get ©Safra,Akavia,Moshkovitz

Lemma’s Proof - Claim 3 By auxiliary lemmas (4||)-1/2  e-2||  (1-2)||, i.e. ||-1/2 (4)1/2 ·(1-2)||, which yields the following bound That is, Now applying claim 2 results the desired lower bound ©Safra,Akavia,Moshkovitz

Lemma’s Proof -Conclusion
General Fourier Analysis facts Lemma’s Proof -Conclusion Multiplicative representation Representation by Fourier Basis We showed that there is an assignment scheme with expected success of at least 42 ,  There exists an assignment that satisfies at least 42 of the tests in par[,k]  (par[,k]) > 42 Q.E.D. Claim 2 Claim 1 Claim 3:The expected success of the distributional assignment  on [C,V]par[,k] is at least 4 2 (par[,k]) > 42  ©Safra,Akavia,Moshkovitz

Home Assignment Show it is NP-hard, for any  > 0, given a 3SAT instance , to distinguish between the case where () = 1, and the case in which () < 7/8+ Hint: Let ’s variables be as in L, and ’s clauses to take the form f OR g OR ‘f* + g + h’ for f and g chosen in the same way as in L, while h is chosen as follows: h(b) = 1 for b such that f(b|V) and g(b) are both FALSE For all other b’s, independently for each b, h(b)=1 with probability , and 0 with probability 1- ©Safra,Akavia,Moshkovitz

Appendix

Expanders Def: a graph G(V,E) is a c-expander if for every SV,
Gap-3-SAT () = ()  Parallel repetition lemma L  LLC-Lemma: (L) = ½+/2  (par[,k]) > 42 expander Long code Gap-3-SAT-7 par[, k] Expanders Def: a graph G(V,E) is a c-expander if for every SV, |S| ½|V|: |N(S)\S|  c·|S| [where N(S) denotes the set of neighbors of S] Lemma: For every m, one can construct in poly-time a 3-regular, m-vertexes, c-expander, for some constant c>0 Corollary: a cut between S and V\S, for |S|  ½|V| must contain > c·|S| edges  ©Safra,Akavia,Moshkovitz

Reduction Using Expanders
Assume ’ for which (’) is either 1 or 1-20/c.  is ’ with the following changes: an occurrence of y in i is replaced by a variable xy,i Let Gy, for every y, be a 3-regular, c-expander over all occurrences xy,i of y For every edge connecting xy,i to xy,j in Gy, add to  the clauses (xy,i  xy,j) and (xy,i   xy,j) It is easy to see that: ||  10 |’| Each variable xy,i of  appears in exactly 7 i   constructible by the Lemma asserting equality ©Safra,Akavia,Moshkovitz

Correctness of the Reduction
Back Correctness of the Reduction  is completely satisfiable iff ’ is In case ’ is unsatisfiable: (’) < 1-20/c Let A be an optimal assignment to  Let Amaj assign xy,i the value assigned by A to the majority, over j, of variables xy,j Let FA and FAmaj be the sets of    unsatisfied by A and Amaj respectively: ||·(1-()) = |FA| = |FAFAmaj|+|FA\FAmaj|  |FAFAmaj|+½c|FAmaj\FA|  ½c|FAmaj| and since Amaj is in fact an assignment to ’ ()  1- ½c(1- (’))/10 < 1- ½c(20/c)/10= 1- ©Safra,Akavia,Moshkovitz

Tight Hardness Results for Some Approximation Problems [Håstad]

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