Corollary . There exists some ρ < 1 such that if there is a P-time ρ -approximation algorithm for Max - 3SAT then P = NP . ◮ The ρ -approximation algorithm for Max - 3SAT is NP-hard. Computational Complexity, by Fu Yuxi PCP Theorem 27 / 119
Equivalence of the Two Views Computational Complexity, by Fu Yuxi PCP Theorem 28 / 119
CSP, Constraint Satisfaction Problem If q is a natural number, then a q CSP instance ϕ with n variables is a collection of constraints ϕ 1 , . . . , ϕ m : { 0 , 1 } n → { 0 , 1 } such that for each i ∈ [ m ] the function ϕ i depends on q of its input locations. We call q the arity of ϕ , and m the size of ϕ . An assignment u ∈ { 0 , 1 } n satisfies a constraint ϕ i if ϕ i ( u ) = 1. Let �� n � i =1 ϕ i ( u ) val ( ϕ ) = max . m u ∈{ 0 , 1 } n We say that ϕ is satisfiable if val ( ϕ ) = 1. q CSP is a generalization of 3 SAT . Computational Complexity, by Fu Yuxi PCP Theorem 29 / 119
1. We assume that n ≤ qm . 2. Since every ϕ i can be described by a formula of size q 2 q , and every variable can be coded up by log n bits, a q CSP instance can be described by O ( mq 2 q log n ) bits. 3. The greedy algorithm for MAX -3 SAT can be applied to MAX q CSP to produce an assignment satisfying ≥ val ( ϕ ) m constraints. 2 q Computational Complexity, by Fu Yuxi PCP Theorem 30 / 119
Gap CSP Suppose q ∈ N and ρ ≤ 1. Let ρ - GAP q CSP be the promise problem of determining if a q CSP instance ϕ satisfies either (1) val ( ϕ ) = 1 or (2) val ( ϕ ) < ρ . We say that ρ - GAP q CSP is NP-hard if for every NP-problem L some P-time computable function f : L → ρ - GAP q CSP exists such that x ∈ L ⇒ val ( f ( x )) = 1 , x / ∈ L ⇒ val ( f ( x )) < ρ. PCP Theorem. There exists some ρ ∈ (0 , 1) such that ρ - GAP 3 SAT is NP-hard. Computational Complexity, by Fu Yuxi PCP Theorem 31 / 119
PCP Theorem . There exist q ∈ N and ρ ∈ (0 , 1) such that ρ - GAP q CSP is NP-hard. Computational Complexity, by Fu Yuxi PCP Theorem 32 / 119
Equivalence Proof PCP Theorem ⇒ PCP Theorem . This is essentially the Cook-Levin reduction. 1. Suppose NP ⊆ PCP (log , 1). Then 3 SAT has a PCP verifier V that makes q queries using c log n random bits. 2. Given input x with | x | = n and random string r ∈ { 0 , 1 } c log n , V ( x , r ) is a Boolean function of type { 0 , 1 } q → { 0 , 1 } . 3. ϕ = { V ( x , r ) } r ∈{ 0 , 1 } c log n is a P-size q CSP instance. ◮ By completeness, x ∈ 3SAT ⇒ val ( ϕ ) = 1. ◮ By soundness, x / ∈ 3SAT ⇒ val ( ϕ ) ≤ 1 / 2. 4. The map from 3 SAT to 1 2 - GAP q CSP is P-time computable. ◮ V runs in P-time. Computational Complexity, by Fu Yuxi PCP Theorem 33 / 119
Equivalence Proof PCP Theorem ⇐ PCP Theorem . Suppose L ∈ NP and ρ - GAP q CSP is NP-hard for some q ∈ N , ρ < 1. By assumption there is some P-time reduction f : L → ρ - GAP q CSP . 1. The verifier for L works as follows: ◮ On input x , compute the q CSP instance f ( x ) = { ϕ i } i ∈ [ m ] . ◮ Given a proof π , which is an assignment to the variables, it randomly chooses i ∈ [ m ] and checks if ϕ i is satisfied by reading the relevant q bits of the proof. 2. If x ∈ L , the verifier always accepts; otherwise it accepts with probability < ρ . Computational Complexity, by Fu Yuxi PCP Theorem 34 / 119
Equivalence Proof PCP Theorem ⇔ PCP Theorem . This is very much like the equivalence between SAT and 3 SAT . 1. Let ǫ > 0 and q ∈ N be such that (1 − ǫ )- GAP q CSP is NP-hard. 2. Let ϕ = { ϕ i } m i =1 be a q CSP instance with n variables. 3. Each ϕ i is the conjunction of at most 2 q clauses, each being the disjunction of at most q literals. 4. If all assignments fail at least an ǫ fragment of the constraints of ϕ , then all ǫ assignments fail at least a q 2 q fragment of the clauses of the 3 SAT instance. Computational Complexity, by Fu Yuxi PCP Theorem 35 / 119
Proof View Inapproximability View PCP verifier V CSP instance ϕ PCP proof π assignment to variables u proof length | π | number of variables n number of queries q arity of constraints q number of random bits r logarithm of number of constraints log m soundness parameter ǫ maximum fraction of violated constraints of no instance NP ⊆ PCP (log , 1) ρ - GAP q CSP is NP-hard The equivalence of the proof view and the inapproximability view is essentially due to the Cook-Levin Theorem for PTM. Computational Complexity, by Fu Yuxi PCP Theorem 36 / 119
Inapproximability Computational Complexity, by Fu Yuxi PCP Theorem 37 / 119
Min - VC and Max - IS are inherently different from the perspective of approximation. ◮ Min - VC + Max - IS = n . n − IS ◮ ρ -approximation algorithm of Max - IS ⇒ n − ρ IS -approximation algorithm of Min - VC . Computational Complexity, by Fu Yuxi PCP Theorem 38 / 119
Lemma . There is a P-time computable function f from 3CNF formulas to graphs that maps a formula ϕ to an n -vertex graph f ( ϕ ) whose independent set is of size val ( ϕ ) n 7 . The standard Karp reduction from 3SAT to Max - IS is as follows: ◮ Each clause is translated to a clique of 7 nodes, each node represents a (partial) assignment that validates the clause. ◮ Two nodes from two different cliques are connected if and only if they conflict. A 3CNF formula ϕ consisting of m clauses is translated to a graph with 7 m nodes, and an assignment satisfying l clauses of ϕ if and only if the graph has an independent set of size l . Computational Complexity, by Fu Yuxi PCP Theorem 39 / 119
Theorem . The following statements are valid. 1. ∃ ρ ′ < 1. ρ ′ -approximation to Min - VC is NP-hard, and 2. ∀ ρ < 1. ρ -approximation to Max - IS is NP-hard. [ ∃ ρ . ρ -approximation to Max - IS is NP-hard.] ∗ By PCP Theorem, ρ -approximation to Max - 3SAT is NP-hard for some ρ . So by Lemma ρ -approximation to Max - IS is NP-hard. 1. Referring to the map of Lemma, the minimum vertex cover has size n − val ( ϕ ) n 7 . Let ρ ′ = 7 − ρ . Suppose Min - VC had a ρ ′ -approximation algorithm. 6 ◮ If val ( ϕ ) = 1, it would produce a vertex cover of size ≤ 1 ρ ′ ( n − n 7 ) = n − ρ n 7 . ◮ If val ( ϕ ) < ρ , the minimum vertex cover has size > n − ρ n 7 . The algorithm must return a vertex cover of size > n − ρ n 7 . The first proposition is established. The second will be proved by making use of [ ] ∗ . Computational Complexity, by Fu Yuxi PCP Theorem 40 / 119
� IS � � ρ IS � 2. Assume that Max - IS were ρ 0 -approximate. Let k satisfy ρ 0 > . k k 1. Suppose G is the input graph. Construct G k as follows: ◮ The vertices are k -size subsets of V G ; ◮ Two vertices S 1 , S 2 are disconnected if S 1 ∪ S 2 is an independent set of G . 2. Apply the ρ 0 -approximation algorithm to G k , and derive an independent set of G from the output of the algorithm. � IS � ◮ The largest independent set of G k is of size , where IS is the maximum k independent set of G . � IS � � ρ IS � ◮ The output is an independent set of size ≥ ρ 0 > . k k ◮ An independent set of size ρ IS can be derived. This is a contradiction. Computational Complexity, by Fu Yuxi PCP Theorem 41 / 119
The Fourier Transform Technique Computational Complexity, by Fu Yuxi PCP Theorem 42 / 119
A Boolean function f : { 0 , 1 } n → { 0 , 1 } is a linear function if f ( x + y ) = f ( x ) + f ( y ) , where “+” is the exclusive-or operator. Computational Complexity, by Fu Yuxi PCP Theorem 43 / 119
Fourier Transform over GF (2) n Boolean functions can be studied using Fourier transform over GF (2) n . We shall use { +1 , − 1 } instead of { 0 , 1 } in this section. ◮ 0 ↔ ( − 1) 0 = 1 and 1 ↔ ( − 1) 1 = − 1. ◮ { 0 , 1 } n is turned into {± 1 } n . ◮ ”exclusive-or” is turned into “multiplication”. Computational Complexity, by Fu Yuxi PCP Theorem 44 / 119
Fourier Transform over GF (2) n The 2 n -dimensional Hilbert space R {± 1 } n is defined as follows: For f , g ∈ R {± 1 } n , 1. ( f + g )( x ) = f ( x ) + g ( x ), 2. ( cf )( x ) = cf ( x ), and 3. expectation inner product: � f , g � = E x ∈{± 1 } n [ f ( x ) g ( x )]. Standard orthogonal basis: { e x } x ∈{± 1 } n . Computational Complexity, by Fu Yuxi PCP Theorem 45 / 119
Fourier Transform over GF (2) n Fourier Basis: { χ α } α ⊆ [ n ] , where χ α ( x ) = � i ∈ α x i . 1. χ ∅ = 1 . 2. We will see that Fourier basis functions are the same as the linear functions. 3. Inner product � χ α , χ β � = 1 if α = β and = 0 otherwise. Fourier basis is orthonormal. ◮ � χ α , χ α � = E x ∈{± 1 } n [ χ α ( x ) χ α ( x )] = E x ∈{± 1 } n [ χ α ( x · x )] = E x ∈{± 1 } n [1] = 1. ◮ � χ α , χ β � = E x ∈{± 1 } n [ χ α ( x ) χ β ( x )] = 0 if α � = β . The notation y · z stands for the dot product of y and z . Computational Complexity, by Fu Yuxi PCP Theorem 46 / 119
Fourier Transform over GF (2) n f = � α ⊆ [ n ] � f α χ α for every f ∈ R {± 1 } n , where � f α is the α th Fourier coefficient of f . Lemma . (i) � f , g � = � g α . (ii) (Parsevals Identity) � f , f � = � α ⊆ [ n ] � α ⊆ [ n ] � f 2 f α � α . Proof. � f , g � = � � f α χ α , � g β χ β � = � g α � χ α , χ β � = � α ⊆ [ n ] � α,β ⊆ [ n ] � α ⊆ [ n ] � β ⊆ [ n ] � f α � f α � g α . Computational Complexity, by Fu Yuxi PCP Theorem 47 / 119
Fourier Transform over GF (2) n Example. 1. Majority function of 3 variables = 1 2 u 1 + 1 2 u 2 + 1 2 u 3 − 1 2 u 1 u 2 u 3 . 2. Projection function λ x 1 . . . x n . x i . Here � f α is 1 if α = { i } and is 0 if α � = { i } . Computational Complexity, by Fu Yuxi PCP Theorem 48 / 119
Fourier Transform over GF (2) n Theorem . Suppose f : {± 1 } n → {± 1 } satisfies Pr x , y [ f ( x · y ) = f ( x ) f ( y )] ≥ 1 2 + ǫ . Then there is some α ⊆ [ n ] such that � f α ≥ 2 ǫ . The assumption is equivalent to E x , y [ f ( x · y ) f ( x ) f ( y )] ≥ 1 2 + ǫ − ( 1 2 − ǫ ) = 2 ǫ . Now � � � � � � 2 ǫ ≤ E x , y [ f ( x · y ) f ( x ) f ( y )] = E x , y [( f α χ α ( x · y ))( f β χ β ( x ))( f γ χ γ ( y ))] α β γ � f α � � f β � = E x , y [ f γ χ α ( x ) χ α ( y ) χ β ( x ) χ γ ( y )] α,β γ � � f α � f β � = f γ E x , y [ χ α ( x ) χ α ( y ) χ β ( x ) χ γ ( y )] α,β,γ � � f α � f β � = f γ E x [ χ α ( x ) χ β ( x )] E y [ χ α ( y ) χ γ ( y )] α,β,γ � � f 3 � � � f 2 � = α ≤ (max f α ) α = max f α . α α α α The last equality is due to the fact that f is Boolean. Computational Complexity, by Fu Yuxi PCP Theorem 49 / 119
Fourier Transform over GF (2) n For Boolean functions f , g : {± 1 } n → {± 1 } , the inner product � f , g � is |{ x | f ( x ) = g ( x ) }| − |{ x | f ( x ) � = g ( x ) }| . 2 n 2 n Referring to the Theorem, let � f α ≥ 2 ǫ be the largest Fourier coefficient of f . Then � f , χ α � ≥ 2 ǫ. In other words f coincides with the basis function χ α on ≥ 1 2 + ǫ fraction of inputs. Computational Complexity, by Fu Yuxi PCP Theorem 50 / 119
Efficient Conversion of NP Certificate to PCP Proof Computational Complexity, by Fu Yuxi PCP Theorem 51 / 119
Proofs of PCP Theorems involve some interesting ways of encoding NP-certificates and the associated methods of checking if a string is a valid encoding. One idea is to amplify any error that appears in an NP-certificate. We shall showcase how it works by looking at a problem to which the amplification power of Walsh-Hadamard Code can be exploited. Theorem . NP ⊆ PCP ( poly ( n ) , 1). Computational Complexity, by Fu Yuxi PCP Theorem 52 / 119
Walsh-Hadamard Code The Walsh-Hadamard function WH : { 0 , 1 } n → { 0 , 1 } 2 n encodes a string of length n by a function in n variables over GF (2): WH ( u ) : x �→ u ⊙ x , where u ⊙ x = � n i =1 u i x i (mod 2). Random Subsum Principle. ◮ If u � = v then for exactly half the choices of x , u ⊙ x � = v ⊙ x . Computational Complexity, by Fu Yuxi PCP Theorem 53 / 119
Walsh-Hadamard Codeword We say that f is a Walsh-Hadamard codeword if f = WH ( u ) for some u ∈ { 0 , 1 } n . 1. Walsh-Hadamard codewords are precisely the linear functions. This is because a linear function f is the same as WH ( f ), where f ( e 1 ) f ( e 2 ) f = . . . . f ( e n ) 2. Walsh-Hadamard codewords are essentially the Fourier basis functions. Computational Complexity, by Fu Yuxi PCP Theorem 54 / 119
Let ρ ∈ [0 , 1]. The functions f , g : { 0 , 1 } n → { 0 , 1 } are ρ -close if Pr x ∈ R { 0 , 1 } n [ f ( x ) = g ( x )] ≥ ρ. Theorem . Let f : { 0 , 1 } n → { 0 , 1 } be such that for some ρ > 1 2 , Pr x , y ∈ R { 0 , 1 } n [ f ( x + y ) = f ( x ) + f ( y )] ≥ ρ. Then f is ρ -close to a linear function. Proof. Let ρ = 1 2 + ǫ . This follows immediately from the theorem of the previous section. Computational Complexity, by Fu Yuxi PCP Theorem 55 / 119
Local Testing of Walsh-Hadamard Codeword A local test of f checks if f is a Walsh-Hadamard codeword by making a constant number of queries. ◮ It accepts every linear function, and ◮ it rejects every function that is far from being linear with high probability. For δ ∈ (0 , 1 / 2) a (1 − δ )-linearity test rejects with probability ≥ 1 2 any function not (1 − δ )-close to a linear function by testing f ( x + y ) = f ( x ) + f ( y ) randomly for 1 1 δ times. δ ≈ 1 e < 1 The error probability is ≤ (1 − δ ) 2 . Computational Complexity, by Fu Yuxi PCP Theorem 56 / 119
Local Decoding 4 and f is (1 − δ )-close to some linear function � Suppose δ < 1 f . Given x one can learn � f ( x ) by making only two queries to f . 1. Choose x ′ ∈ R { 0 , 1 } n ; 2. Set x ′′ = x + x ′ ; 3. Output f ( x ′ ) + f ( x ′′ ). By union bound � f ( x ) = � f ( x ′ ) + � f ( x ′′ ) = f ( x ′ ) + f ( x ′′ ) holds with probability ≥ 1 − 2 δ . Computational Complexity, by Fu Yuxi PCP Theorem 57 / 119
Quadratic Equation in GF (2) Suppose A is an m × n 2 matrix and b is an m -dimensional vector with values in GF (2). Let ( A , b ) ∈ QUADEQ if there is an n -dimensional vector u such that A ( u ⊗ u ) = b , where u ⊗ u is the tensor product of u . u ⊗ u = ( u 1 u 1 , . . . , u 1 u n , . . . , u n u 1 , . . . , u n u n ) † . Computational Complexity, by Fu Yuxi PCP Theorem 58 / 119
Quadratic Equation in GF (2) An instance of QUADEQ over u 1 , u 2 , u 3 , u 4 , u 5 : u 1 u 2 + u 3 u 4 + u 1 u 5 = 1 u 1 u 1 + u 2 u 3 + u 1 u 4 = 0 A satisfying assignment is (0 , 0 , 1 , 1 , 0). Computational Complexity, by Fu Yuxi PCP Theorem 59 / 119
QUADEQ is NP-Complete CKT - SAT ≤ K QUADEQ . ◮ The inputs and the outputs are turned into variables. ◮ Boolean equality x ∨ y = z relating the inputs to the output is turned into algebraic equality (1 − x )(1 − y ) = 1 − z in GF (2), which is equivalent to xx + yy + xy + zz = 0. ◮ ¬ x = z is turned into xx + zz = 1. ◮ x ∧ y = z is turned into xy + zz = 0. Computational Complexity, by Fu Yuxi PCP Theorem 60 / 119
From NP Certificate to PCP Proof A certificate for ( A , b ) is an n -dimensional vector u witnessing ( A , b ) ∈ QUADEQ . ◮ To check if u is a solution, one reads the n bits of u and checks the m equations. We convert an NP-certificate u to the PCP-proof WH ( u ) WH ( u ⊗ u ). ◮ The proof is a string of length 2 n + 2 n 2 . ◮ Using the proof it is straightforward to verify probabilistically if ( A , b ) ∈ QUADEQ . Computational Complexity, by Fu Yuxi PCP Theorem 61 / 119
Verifier for QUADEQ Step 1. Verify that f , g are linear functions. 1. Perform a 0 . 999-linearity test on f , g . If successful we may assume that f ( r ) = u ⊙ r and g ( z ) = w ⊙ z . Computational Complexity, by Fu Yuxi PCP Theorem 62 / 119
Verifier for QUADEQ Step 2. Verify that g encodes ( u ⊗ u ) ⊙ . 1. Get independent random r , r ′ . 2. Reject if f ( r ) f ( r ′ ) � = g ( r ⊗ r ′ ). 3. Repeat the test 10 times. ◮ In a correct proof f ( r ) f ( r ′ ) = ( � i u i r i )( � j ) = � j u j r ′ i , j u i u j r i r ′ j = g ( r ⊗ r ′ ). ◮ Assume w � = u ⊗ u . Let matrices W and U be w and respectively u ⊗ u . One has ◮ g ( r ⊗ r ′ ) = w ⊙ ( r ⊗ r ′ ) = � i , j w ij r i r ′ j = r W r ′ , and ◮ f ( r ) f ( r ′ ) = ( u ⊗ r )( u ⊗ r ′ ) = ( � i u i r i )( � j u j r ′ j ) = r U r ′ . 2 of r ’s; and r W r ′ , r U r ′ differ for at least 1 r W , r U differ for at least 1 4 of ( r , r ′ )’s. 4 ) 10 > 0 . 9. ◮ The overall probability of rejection is at least 1 − ( 3 Computational Complexity, by Fu Yuxi PCP Theorem 63 / 119
Verifier for QUADEQ Step 3. Verify that g encodes a solution. 1. Take a random subset S of [ m ]. 2. Reject if g ( � k ∈ S A k , ) � = � k ∈ S b k . ◮ There is enough time to check A ( u ⊗ u ) = b by checking, for every k ∈ [ m ], the equality g ( A k , ) = b k . ◮ However since m is part of the input, the number of queries, which must be a constant, should not depend on m . ◮ If { k ∈ [ m ] | g ( A k , ) � = b k } � = ∅ , then Pr S ⊆ R [ m ] [ | S ∩ { k ∈ [ m ] | g ( A k , ) � = b k }| is odd ] = 1 2 . Note that g ( � k ∈ S A k , ) = � k ∈ S g ( A k , ) by linearity. Computational Complexity, by Fu Yuxi PCP Theorem 64 / 119
Suppose the PCP verifier for QUADEQ makes q 0 queries. It follows from the completeness of QUADEQ that all NP problems have PCP verifiers that toss coins for a polynomial number of time and make precisely q 0 queries. Computational Complexity, by Fu Yuxi PCP Theorem 65 / 119
Proof of PCP Theorem Computational Complexity, by Fu Yuxi PCP Theorem 66 / 119
CSP with Nonbinary Alphabet q CSP W is analogous to q CSP except that the alphabet is [ W ] instead of { 0 , 1 } . The constraints are functions of type [ W ] q → { 0 , 1 } . For ρ ∈ (0 , 1), we define the promise problem ρ - GAP q CSP W analogous to ρ - GAP q CSP . Computational Complexity, by Fu Yuxi PCP Theorem 67 / 119
3 COL is a case of 2 CSP 3 . Computational Complexity, by Fu Yuxi PCP Theorem 68 / 119
PCP Theorem states that ρ - GAP q CSP is NP-hard for some q , ρ . The proof we shall describe is based on the following: 1. If ϕ of m constraints is unsatisfied, then val ( ϕ ) ≤ 1 − 1 m . 2. There is a construction that increases the gap. The idea is to start with an NP-problem, then apply Step 2 for log m times. Computational Complexity, by Fu Yuxi PCP Theorem 69 / 119
Let f be a function mapping CSP instances to CSP instances. It is a CL-reduction (complete linear-blowup reduction) if it is P-time computable and the following are valid for every CSP instance ϕ . ◮ Completeness. If ϕ is satisfiable then f ( ϕ ) is satisfiable. ◮ Linear Blowup. If ϕ has m constraints, f ( ϕ ) has no more than Cm constraints over a new alphabet W . ◮ C , W depend on neither the number of constraints nor the number of variables of ϕ . Computational Complexity, by Fu Yuxi PCP Theorem 70 / 119
Main Lemma . There exist constants q 0 ≥ 3, ǫ 0 > 0 and CL-reduction f such that for every q 0 CSP instance ϕ and every ǫ < ǫ 0 , f ( ϕ ) is a q 0 CSP instance satisfying val ( ϕ ) ≤ 1 − ǫ ⇒ val ( f ( ϕ )) ≤ 1 − 2 ǫ. q 0 CSP Instance Arity Alphabet Constraint Gap 1 − ǫ ϕ q 0 binary m ⇓ ⇓ ⇓ ⇓ ⇓ 1 − 2 ǫ f ( ϕ ) q 0 binary Cm In the following proof of the PCP Theorem, q 0 , ǫ 0 , ℓ , W , W ′ , d , t are all constants. They depend neither on the number n of variables nor on the number m of constraints. Computational Complexity, by Fu Yuxi PCP Theorem 71 / 119
Proof of PCP Theorem Let q 0 ≥ 3 and ǫ 0 > 0 be given by the Main Lemma. A CL-reduction from q 0 CSP to (1 − 2 ǫ 0 )- GAP q 0 CSP is obtained as follows: 1. q 0 CSP is NP-hard. 2. For a q 0 CSP instance ϕ with m constraints, apply Main Lemma for log m times to amplify the gap. We get an instance ψ . 3. If ϕ is satisfiable, then ψ is satisfiable. Otherwise according to Main Lemma val ( ψ ) ≤ 1 − 2 log m · 1 m ≤ 1 − 2 ǫ 0 . 4. | ψ | ≤ C log m m = poly ( | ϕ | ). Conclude that (1 − 2 ǫ 0 )- GAP q 0 CSP is NP-hard. C depends on two constants, q 0 and 2 (the size of alphabet). Computational Complexity, by Fu Yuxi PCP Theorem 72 / 119
Main Lemma is proved in three steps. 1. Prove that every q CSP instance can be turned into a “nice” q CSP W instance. 2. Gap Amplification. Construct a CL-reduction f that increases both the gap and the alphabet size of a “nice” q CSP instance. [Dinur’s proof] 3. Alphabet Reduction. Construct a CL-reduction g that decreases the alphabet size to 2 with a modest reduction in the gap. [Proof of Arora et al.] Computational Complexity, by Fu Yuxi PCP Theorem 73 / 119
Gap Amplification . For all numbers ℓ, q , there are number W , ǫ 0 ∈ (0 , 1) and a CL-reduction g ℓ, q such that for every q CSP instance ϕ , ψ = g ℓ, q ( ϕ ) is a 2 CSP W instance that satisfies the following for all ǫ < ǫ 0 . val ( ϕ ) ≤ 1 − ǫ ⇒ val ( ψ ) ≤ 1 − ℓǫ. Alphabet Reduction . There exist a constant q 0 and a CL-reduction h such that for every 2 CSP W instance ϕ , ψ = h ( ϕ ) is a q 0 CSP instance satisfying val ( ϕ ) ≤ 1 − ǫ ⇒ val ( ψ ) ≤ 1 − ǫ/ 3 . CSP Instance Arity Alphabet Constraint Gap ϕ q 0 binary m 1 − ǫ ⇓ ⇓ ⇓ ⇓ ⇓ C ′ m f ( ϕ ) 2 nonbinary 1 − 6 ǫ ⇓ ⇓ ⇓ ⇓ ⇓ C ′′ C ′ m g ( f ( ϕ )) q 0 binary 1 − 2 ǫ Computational Complexity, by Fu Yuxi PCP Theorem 74 / 119
Dinur makes use of expander graphs to construct new constraints. Let ϕ be a 2 CSP W instance with n variables. The constraint graph G ϕ of ϕ is defined as follows: 1. the vertex set is [ n ], and 2. ( i , j ) is an edge if there is a constraint on the variables u i , u j . Parallel edges and self-loops are allowed. Computational Complexity, by Fu Yuxi PCP Theorem 75 / 119
A 2 CSP W instance ϕ is nice if the following are valid: 1. There is a constant d such that G ϕ is a ( d , 0 . 9)-expander. 2. At every vertex half of the adjacent edges are self loops. A nice CSP instance looks like an expander. In a nice CSP a t + 1 step random walk is very much like a t step random walk. Computational Complexity, by Fu Yuxi PCP Theorem 76 / 119
Lemma . Let G be a d -regular n -vertex graph, S be a vertex subset and T = S . Then | E ( S , T ) | ≥ (1 − λ G ) d | S || T | | S | + | T | . (1) The vector x defined below satisfies � x � 2 2 = | S || T | ( | S | + | T | ) and x ⊥ 1 . � + | T | , i ∈ S , x i = −| S | , i ∈ T . Let Z = � i , j A i , j ( x i − x j ) 2 . By definition Z = 2 d | E ( S , T ) | ( | S | + | T | ) 2 . On the other hand � � � A i , j x 2 A i , j x 2 j = 2 � x � 2 Z = i − 2 A i , j x i x j + 2 − 2 � x , A x � . i , j i , j i , j Since x ⊥ 1 , � x , A x � ≤ λ G � x � 2 2 (cf. Rayleigh quotient). Computational Complexity, by Fu Yuxi PCP Theorem 77 / 119
Let G = ( V , E ) be an expander and S ⊆ V with | S | ≤ | V | / 2. The following holds. � 1 � Pr ( u , v ) ∈ E [ u ∈ S , v ∈ S ] ≤ | S | 2 + λ G . (2) | V | 2 Observe that | S | / | V | = Pr ( u , v ) ∈ E [ u ∈ S , v ∈ S ] + Pr ( u , v ) ∈ E [ u ∈ S , v ∈ S ]. And by (1), one has Pr ( u , v ) ∈ E [ u ∈ S , v ∈ S ] = E ( S , S ) / d | V | ≥ | S | | V | · 1 2 · (1 − λ G ) . We are done by substituting | S | / | V | − Pr ( u , v ) ∈ E [ u ∈ S , v ∈ S ] for Pr ( u , v ) ∈ E [ u ∈ S , v ∈ S ]. � 1 � 2 + λ ℓ Pr ( u , v ) ∈ E ℓ [ u ∈ S , v ∈ S ] ≤ | S | G . (3) | V | 2 Computational Complexity, by Fu Yuxi PCP Theorem 78 / 119
Step 1: Reduction to Nice Instance The reduction consists of three steps. Step1 . 1 ⇒ 2 CSP 2 q 0 instance q 0 CSP instance Step1 . 2 ⇒ 2 CSP 2 q 0 instance with regular constraint graph Step1 . 3 ⇒ nice 2 CSP 2 q 0 instance . In all the three steps the fraction of violated constraints decreases. Computational Complexity, by Fu Yuxi PCP Theorem 79 / 119
Step 1: Reduction to Nice Instance Step 1.1 . There exists a CL-reduction that maps a q 0 CSP instance ϕ to a 2 CSP 2 q 0 instance ψ such that val ( ϕ ) ≤ 1 − ǫ ⇒ val ( ψ ) ≤ 1 − ǫ . q 0 Suppose ϕ has variables x 1 , . . . , x n and m constraints. ◮ The new instance ψ has variables x 1 , . . . , x n , y 1 , . . . , y m , where y i ∈ { 0 , 1 } q 0 codes up an assignment to x 1 , . . . , x n . ◮ For each variable x j in ϕ i , construct the constraint ψ i , j stating that y i satisfies ϕ i and y i is consistent to x j . What is the formula ψ ij ( y i , x j )? What is the size of ψ ij ( y i , x j )? Computational Complexity, by Fu Yuxi PCP Theorem 80 / 119
Step 1: Reduction to Nice Instance Step 1.2 . There exist an absolute constant d and a CL-reduction that maps a 2 CSP W instance ϕ to a 2 CSP W instance ψ such that ǫ val ( ϕ ) ≤ 1 − ǫ ⇒ val ( ψ ) ≤ 1 − 100 Wd and that G ψ is d -regular. Let { G k } k be an explicit ( d − 1 , 0 . 9)-expander. We get ψ by replacing each k -degree i = y j ′ node of G ϕ by G k and adding the identity constraint (of the form y j i , where i ∈ [ n ]) to each edge ( j , j ′ ) of G k . If ϕ has m constraints, ψ has dm constraints. Suppose val ( ϕ ) ≤ 1 − ǫ and v is an unsatisfying assignment to ψ . ǫ m It suffices to prove that v violates at least 100 W constraints of ψ . Continue on the next slide. Fact . For every c ∈ (0 , 1) there is a constant d and an algorithm that, given input n , runs in poly ( n ) time and outputs an ( n , d , c )-expander. Computational Complexity, by Fu Yuxi PCP Theorem 81 / 119
Step 1: Reduction to Nice Instance Let u be the assignment to ϕ that is defined by the plurality of the assignment v to ψ . Let t i be the number of v j i ’s, where j ∈ [ k ], that disagree with u i . Clearly t i ≤ k (1 − 1 W ). If � n i =1 t i is large, each G k already contains enough violated constraints. 1. � n 4 ǫ m . Let S i = { y j i | v j i =1 t i ≥ 1 i = u i } and let S i = { y 1 i , . . . , y k i } \ S i . The number of constraints of G k violated by v is at least (1) ≥ (1 − λ G k )( d − 1) | S i || S i | = 1 d − 1 1 E ( S i , S i ) | S i || S i | ≥ 10 W t i , 10 k | S i | + | S i | where ≥ is due to | S i | ≥ k / W . Now � ǫ m ǫ i ∈ [ n ] E ( S i , S i ) ≥ 40 Wd · dm . 40 W = 2. � n i =1 t i < 1 4 ǫ m . Since val ( ϕ ) ≤ 1 − ǫ , there is at least ǫ m constraints violated in ϕ by u . These ǫ m constraints are also in ψ with variables being v . Since every constraint has two variables, less than 1 4 ǫ m + 1 4 ǫ m constraints have valuations in ψ different from those in ϕ . So at least 1 2 ǫ m constraints are violated. Computational Complexity, by Fu Yuxi PCP Theorem 82 / 119
Step 1: Reduction to Nice Instance Step 1.3 . There is an absolute constant d and a CL-reduction that maps a 2 CSP W instance ϕ with G ϕ being d ′ -regular for some d ′ ≤ d to a 2 CSP W instance ψ such that ǫ val ( ϕ ) ≤ 1 − ǫ ⇒ val ( ψ ) < 1 − 10 d and that G ψ is nice, 4 d -regular, and half of the edges adjacent to each vertex are loops. There is a constant d and an explicit ( d , 0 . 1)-expander { G n } n ∈ N . We may assume that ϕ is d -regular (adding self-loops if d ′ < d ) and that ϕ contains n variables. We get ψ from ϕ by adding a tautological constraint for every edge of G n and adding 2 d tautological constraints forming self-loops for each vertex. Then λ G ψ ≤ 3 4 + 1 4 λ G ϕ < 0 . 9 . Notice that “adding” decreases ǫ by a factor ≤ d and “adding” by a further factor ≤ 4. Computational Complexity, by Fu Yuxi PCP Theorem 83 / 119
Step 2: Gap Amplification To amplify the gap, we apply a path-product like operation on constraint graphs so that a new constraint is a conjunction of some old constraints. Path construction increases the fraction of violated constraints. We need to address the issue of how to compose two-variable constraints so that every new constraint contains two variables. Computational Complexity, by Fu Yuxi PCP Theorem 84 / 119
Step 2: Gap Amplification Construction of the 2 CSP W ′ instance ψ t : Variables. 1. Let x 1 , . . . , x n denote the variables of ψ . 2. ψ t contains n variables y 1 , . . . , y n ∈ W ′ < W d 5 t , where y i is an assignment to those of x 1 , . . . , x n reachable within t + δ ( t ) steps from x i . We will let δ ( t ) = √ t . For i , j ∈ [ n ] we say that an assignment to y i claims a value for x j . The belt zone is for consistence checking. x j √ t t + t x i x k Computational Complexity, by Fu Yuxi PCP Theorem 85 / 119
Step 2: Gap Amplification Construction of the 2 CSP W ′ instance ψ t : Constraints. 1. For each 2 t +1 step path p = ( i 1 , . . . , i 2 t +2 ) in G ψ , introduce C p = � j ∈ [2 t +2] C j p as a constraint of ψ t such that for each j that C j p appears in C p , the following hold: 1.1 The length of ( i 1 , . . . , i j ) and the length of ( i j +1 , . . . , i 2 t +2 ) are ≤ t + √ t ; 1.2 C j p is obtained from the constraint of x i j and x i j +1 by replacing x i j , x i j +1 respectively by y i 1 ’s claim for x i j and y i 2 t +2 ’s claim for x i j +1 . i j i j +1 √ √ t + t i 1 i 2 t +2 t + t t t Computational Complexity, by Fu Yuxi PCP Theorem 86 / 119
Step 2: Gap Amplification Lemma . There is an algorithm that given t > 1 and a nice 2 CSP W instance ψ with n 2 edges, d -degree G ψ , produces a 2 CSP W ′ instance ψ t satisfying 1-4. variables, m = dn 1. W ′ < W d 5 t and ψ t has at most n · d t + √ t +1 constraints. 2. If ψ is satisfiable then ψ t is satisfiable. √ t d √ t , if val ( ψ ) ≤ 1 − ǫ , then val ( ψ t ) ≤ 1 − ℓǫ for ℓ = 1 3. For ǫ < 10 4 dW 5 . 4. The formula ψ t is produced in poly ( m , W d 5 t ) time. Gap Amplification follows immediately from the lemma. ◮ If ℓ = 6 and W = 2 q 0 , we get a constant t and a constant ǫ 0 = 1 d √ t . ◮ In this case ψ t has O ( m ) constraints and is produced in poly ( m ) time. Computational Complexity, by Fu Yuxi PCP Theorem 87 / 119
Step 2: Gap Amplification The conditions 1, 2, 4 of the lemma are satisfied. Computational Complexity, by Fu Yuxi PCP Theorem 88 / 119
Step 2: Gap Amplification Fix an arbitrary assignment v 1 , . . . , v n to y 1 , . . . , y n . We would like to define an assignment to x 1 , . . . , x n from v 1 , . . . , v n . 1. Let Z i ∈ [ W ] be a random variable defined by the following. ◮ Starting from the vertex i , take a t step random walk in G ψ to reach some vertex k ; output v k ’s claim for x i . Let w i denote the most likely value of Z i . 1 2. We call w 1 , . . . , w n the plurality assignment to x 1 , . . . , x n . Clearly Pr [ Z i = w i ] ≥ W . You are what your friends think you are. Computational Complexity, by Fu Yuxi PCP Theorem 89 / 119
Step 2: Gap Amplification 2 A random 2 t +1 step path p = ( i 1 , . . . , i 2 t +2 ) is picked up with probability n · d 2 t +1 . Equivalently such a random path can be chosen in either of the following manners. 1. Pick up a random node, and take a 2 t +1 step random walk from the node. 2. Pick up a random edge, and then take a j -step random walk from one node of the edge and a (2 t − j )-step random walk from the other node of the edge. Pick up a random constraint of ψ t is the same as picking up a random 2 t +1 step path. Computational Complexity, by Fu Yuxi PCP Theorem 90 / 119
Step 2: Gap Amplification For j ∈ [2 t +1] the j -th edge ( i j , i j +1 ) in p is truthful if v i 1 claims the plurality value for i j and v i 2 t +2 claims the plurality value for i j +1 . Suppose val ( ψ ) ≤ 1 − ǫ . There is a set F of ǫ m = ǫ dn 2 constraints in ψ violated by the assignment x 1 = w 1 , . . . , x n = w n ◮ If p has an edge that is both truthful and in F , the constraint C p is violated. 100 W . For each j ∈ [ t + 1 , t + δ √ t ], 1 Claim . Let δ = the proof is on the next slide 1 Pr p Pr e ∈ R E [ the j th edge of p is truthful | the j th edge of p is e ] ≥ 2 W 2 . ǫ Consequently an edge in p is violated with probability at least 2 W 2 . Computational Complexity, by Fu Yuxi PCP Theorem 91 / 119
A 2 t +1 step random walk with ( i j , i j +1 ) = e can be generated by a ( j − 1)-step random walk from i j and a (2 t − j +1)-step random walk from i j +1 . It boils down to proving 1 Pr [ v i 1 claims the plurality value for i j ] · Pr [ v i 2 t +2 claims the plurality value for i j +1 ] ≥ 2 W 2 . (4) If j = t + 1 then the left hand side of (4) is at least 1 / W 2 . Otherwise observe that ◮ a j -step random walk can be generated by tossing coin for j times and taking S j -step non-self-loop random walk, where S j = ♯ head’s, and that ◮ the statistical distance ∆( S t , S t + δ √ t ) is bounded by 10 δ . Intuitively it is very likely that starting from a same vertex a ( t + δ √ t )-step random walk and a t -step walk would end up in the same vertex. � 1 � � 1 � 1 Thus the left hand side of (4) is at least W − 10 δ W − 10 δ ≥ 2 W 2 . Computational Complexity, by Fu Yuxi PCP Theorem 92 / 119
Step 2: Gap Amplification Let V be the random variable for the number of edges among the middle δ √ t edges that are truthful and in F . Pr [ V > 0] is the probability of a ψ t ’s constraint being violated. If Pr [ V > 0] ≥ ǫ ′ , then at least ǫ ′ fraction of ψ t ’s constraints are violated. Lemma . For every non-negative random variable V , Pr [ V > 0] ≥ E [ V ] 2 E [ V 2 ] . Proof. E [ V | V > 0] 2 ≤ E [ V 2 | V > 0] by convex property. The lemma follows from the following. ◮ E [ V 2 | V > 0] = � i i 2 · Pr [ V = i | V > 0] = � i i 2 · Pr [ V = i ] E [ V 2 ] Pr [ V > 0] = Pr [ V > 0] . �� � 2 � � 2 ◮ E [ V | V > 0] 2 = ( � i i · Pr [ V = i | V > 0]) 2 = i i · Pr [ V = i ] E [ V ] = . Pr [ V > 0] Pr [ V > 0] Computational Complexity, by Fu Yuxi PCP Theorem 93 / 119
Step 2: Gap Amplification Claim . E [ V ] ≥ δ √ t ǫ 2 W 2 . Proof. By the previous claim, the probability of an edge in the middle interval of size δ √ t 2 W 2 . Then E [ V ] ≥ δ √ t ǫ ǫ that is truthful and in F is at least 2 W 2 by linearity. � Claim . E [ V 2 ] ≤ 11 δ √ td ǫ . Proof. ◮ Let V ′ be the number of edges in the middle interval that are in F . Now V ≤ V ′ . It suffices to show that E [ V ′ 2 ] ≤ 11 δ √ td ǫ . ◮ For j ∈ { t + 1 , . . . , t + δ √ t } , let I j be an indicator random variable that is 1 if the j th edge is in F and 0 otherwise. Then V ′ = � j ∈{ t +1 ,..., t + δ √ t } I j . ◮ Let S be the set of end points of the edges in F . Then | S | dn ≤ ǫ . continue on next slide. Computational Complexity, by Fu Yuxi PCP Theorem 94 / 119
Step 2: Gap Amplification �� � � E [ V ′ 2 ] I 2 = E + E I j I j ′ j j j � = j ′ √ � Pr [ j th edge is in F ∧ j ′ th edge is in F ] = ǫδ t + 2 j < j ′ � √ Pr [ j th vertex of walk lies in S ∧ j ′ th vertex of walk lies in S ] ≤ t + 2 ǫδ j < j ′ √ � � ≤ ǫδ t + 2 Pr [ j th vertex of walk lies in S ] · Pr ( a , b ) ∈ E j ′− j [ a ∈ S , b ∈ S ] j j ′ > j � � 2 + ( λ G ) j ′ − j √ � � (3) 1 ≤ ǫδ t + 2 d ǫ · d ǫ 2 j j ′ > j � √ √ √ t ) 2 ( d ǫ ) 2 ( λ G ) k 1 + ≤ ǫδ t + ( δ λ G < 0 . 9 and δ < 1 and td ǫ < 1 k ≥ 1 √ ≤ 11 δ td ǫ. Computational Complexity, by Fu Yuxi PCP Theorem 95 / 119
Step 2: Gap Amplification Finally we conclude that E [ V ] 2 Pr [ V > 0] ≥ E [ V 2 ] � δ √ t ǫ � 2 1 ≥ · 11 δ √ td ǫ 2 W 2 √ t = δ · 44 dW 4 · ǫ √ t 1 100 W · 44 dW 4 · ǫ = √ t > 10 4 dW 5 ǫ = ℓǫ. Computational Complexity, by Fu Yuxi PCP Theorem 96 / 119
Step 3: Alphabet Reduction We look for an algorithm that transforms a 2 CSP W to a q 0 CSP instance. A simple idea is to turn a variable over [ W ] to log W boolean variables. ◮ A constraint can be turned into a circuit C of size bounded by 2 2 log W < W 4 . ◮ This would produce a CSP instance of arity 2 log W . The problem with this idea is that 2 log W is greater than q 0 (in fact W ≥ 2 q 0 ). ◮ If we apply Gap Amplification and Alphabet Reduction for log m times, we would get a CSP instance whose arity depends on input size. Computational Complexity, by Fu Yuxi PCP Theorem 97 / 119
Step 3: Alphabet Reduction A more sophisticated idea is to design a PCP checker for constraint checking! 1. We turn the 2 CSP W problem to evaluation checking problem for CKT - SAT . 2. We further turn it to solution checking problem for QUADEQ . 3. We then apply the construction of the PCP verifier (with q 0 queries!) for QUADEQ . 4. Finally we turn the PCP verifier to a q 0 CSP instance. PCP of Proximity, Verifier Composition, Proof Composition. ◮ In some occasions a verifier is allowed to make only a small or constant number of queries. In other words it cannot read any complete assignment to variables. ◮ A solution is to see an assignment as part of a proof. Consequently a verifier can only get to see a fragment of the proof. This is the PCP verifier for QUADEQ ! Computational Complexity, by Fu Yuxi PCP Theorem 98 / 119
Step 3: Alphabet Reduction Suppose a constraint has been converted to a QUADEQ instance. ◮ Let u 1 and u 2 be assignments to log W variables. ◮ Let c be bit string of size ℓ = poly ( W ) that represents the quadratic equations derived from the circuit C . We assume that the first 2 log W bits of c are u 1 u 2 . Let π 1 π 2 π 3 be a PCP proof for the QUADEQ instance, where ◮ π 1 is supposedly WH ( u 1 ), π 2 is supposedly WH ( u 2 ) and π 3 is supposedly WH ( c ). Computational Complexity, by Fu Yuxi PCP Theorem 99 / 119
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