Advanced Algorithms (XIII) Yijia Chen Fudan University

Transcription

1 Advanced Algorithms (XIII) Yijia Chen Fudan University

2 The PCP Theorem Theorem NP = PCP(log n, 1).

3 Motivation

4 Approximate solutions Definition (Approximation of MAX-3SAT) For every 3CNF formula ϕ, the value of ϕ Thus val(ϕ) := # clauses in ϕ satisfied by z max assignment z # clauses in ϕ ϕ is satisfiable val(ϕ) = 1. For every ρ 1, an algorithm A is a ρ-approximation algorithm for MAX-3SAT if for every 3CNF formula ϕ with m clauses, A(ϕ) outputs an assignment satisfying at least ρ val(ϕ) m of ϕ s clauses.

5 1/2-Approximation for MAX-3SAT The algorithm assigns values to the variables one by one in a greedy fashion, whereby the ith variable is assigned the value that results in satisfying at least 1/2 the clauses in which it appears.

6 The minimum vertex cover problem MIN-VERTEX-COVER Input: A graph G = (V, E). Solution: S V with S e for every e E. Cost: S. Goal: max. For ρ 1, a ρ-approximation algorithm for MIN-VERTEX-COVER is an algorithm that on input a graph G outputs a vertex cover whose size is at most 1/ρ times the size of the minimum vertex cover.

7 1/2-Approximation for MIN-VERTEX-COVER 1. Start with S. 2. Pick any edge e = {u, v}. 3. S S {u, v}. 4. Delete u, v, and all edges adjacent to them. 5. If the graph is not empty, then goto 2, else output S.

8 Two Views of the PCP Theorem

9 New, extremely robust proof systems. Approximating combinatorial optimization problems.

10 Let A be any one of the usual axiomatic systems of mathematics for which proofs can be verified by a deterministic TM in time that is polynomial in the length of the proof. Then L = { ϕ, 1 n ϕ has a proof in A of length n }. The PCP Theorem asserts that L has probabilistically checkable certificates: proofs are probabilistically checkable by examining only a constant number of bits in them.

11 A standard definition of NP L NP if there is a polynomial time Turing machine V that, given input x, checks certificates (or membership proofs) to the effect that x L: x L = πv π (x) = 1 x / L = πv π (x) = 0, where V π denotes a verifier with access to certificate π.

12 PCP Definition Let L be a language and q, r : N N. We say that L has an (r(n), q(n))-pcp verifier if there is a polynomial time probabilistic algorithm V satisfying: 1. Efficiency: On input x {0, 1} n and given random access to a string π {0, 1} of length at most q(n)2 r(n), V uses at most r(n) random coins and makes at most q(n) nonadaptive queries to locations of π. 2. Completeness: If x L, then there exists a proof π {0, 1} with Pr [ V π (x) = 1 ] = Soundness: If x / L then for every proof π {0, 1}, Pr [ V π (x) = 1 ] 1/2.

13 PCP ( r(n), q(n) ) Definition L PCP ( r(n), q(n) ) if there are some constants c, d > 0 such that L has a (c r(n), d q(n))-pcp verifier. Theorem NP = PCP(log n, 1).

14 Remarks 1. By the soundness, if x / L then the verifier has to reject every proof with probability at least 1/2, the most difficult part of the proof. 2. Proofs checkable by an (r, q)-verifier are of length at most q2 r, since such a verifier can look on at most this number of locations. 3. PCP ( r(n), q(n) ) ( ) NTIME 2 O(r(n)) q(n). 4. For different NP languages the (O(log n), O(1))-verifiers might use a different constant number of query bits. However, since every NP language is reducible to SAT, all these numbers can be upper bounded by the number of query bits required by a verifier for SAT. 5. The constant 1/2 in the soundness is arbitrary, in the sense that changing it to any other positive constant smaller than 1 will not change the class of languages defined. A PCP verifier with soundness 1/2 that uses r coins and makes q queries can be converted into a PCP verifier using cr coins and cq queries with soundness 2 c by just repeating its execution c times.

15 GNI PCP ( poly(n), 1 ) The graph non-isomorphism problem is defined as GNI = { (G 0, G 1) G0 and G 0 are two non-isomorphic graphs }. Assume G 0 and G 1 are both of n vertices. The proof π contains, for each graph H with n nodes, π(h) {0, 1} corresponding to whether H G 0 or H G 1, otherwise, arbitrary. The verifier randomly picks b {0, 1} and a permutation. She applies the permutation to G b to obtain an isomorphic graph H. Then she accepts iff π(h) = b. If G 0 G 1, then the correct π makes the verifier accept with probability 1. If G 1 G 2, then the probability that any π makes the verifier accept is at most 1/2.

16 Theorem NEXP = PCP ( poly(n), 1 ).

17 PCP and hardness of approximation Theorem PCP Theorem: Hardness of approximation view There exists ρ < 1 such that for every L NP there is a polynomial time function f mapping strings to (representations of) 3CNF formulas such that x L = val(f (x)) = 1 x / L = val(f (x)) < ρ. Corollary There exists some constant ρ < 1 such that if there is a polynomial time ρ-approximation algorithm for MAX-3SAT then P = NP.

18 Equivalence of the Two Views

19 Constraint satisfaction problems (CSP) Definition Let q N. Then a qcsp instance ϕ is a collection of functions ϕ 1,..., ϕ m (i.e., constraints) from {0, 1} n to {0, 1} such that each function ϕ i depends on at most q of its input locations: for every i [m] there exist j 1,..., j q [n] and f : {0, 1} q {0, 1} such that ϕ i (u) = f ( u j1,..., u jq ) for every u {0, 1} n. An assignment u {0, 1} n satisfies ϕ i if ϕ i (u) = 1. Let i [m] val(ϕ) = max ϕ i(m). u m ϕ is satisfiable if val(ϕ) = 1. We call q the arity of ϕ.

20 Remarks 1. The size of a qcsp-instance is the number of constraints it has. 2. We can always assume n qm. 3. A qcsp instance over n variables with m constraints can be described by a binary string of length O ( mq2 q log n ). 4. The simple greedy approximation algorithm for 3SAT can be generalized for the MAXqCSP problem of maximizing the number of satisfied constraints in a given qcsp instance. For any qcsp instance ϕ with m constraints, this algorithm will output an assignment satisfying val(ϕ)m/2 q constraints.

21 Gap CSP Definition Let q N and ρ 1. Then the ρ-gapqcsp is the problem of determining for a given qcsp-instance ϕ whether: 1. val(ϕ) = 1 (ϕ is a YES instance of ρ-gapqcsp), or 2. val(ϕ) < ρ (ϕ is a NO instance). ρ-gapqcsp is NP-hard if there is a polynomial time function f mapping strings to (representations of) qcsp instances satisfying: 1. Completeness: x L = val(f (x)) = Soundness: x / L = val(f (x)) ρ. Theorem There exist constants q N and ρ R with 0 < ρ < 1 such that ρ-gapqcsp is NP-hard.

22 Equivalence of the two views Theorem The following are equivalent. 1. NP = PCP(log n, 1). 2. There exist constants q N and ρ R with 0 < ρ < 1 such that ρ-gapqcsp is NP-hard.

23 PCP = GAP From NP PCP(log n, 1) we show 1/2-GAPqCSP is NP-hard for some q N. 3SAT has a verifier V which makes q queries and uses c log n random coins. Let x {0, 1} n and r {0, 1} c log n. Then V x,r is the function that on input π outputs 1 if V accepts π on input x and coins r. V x,r depends on at most q locations. Thus for every x {0, 1} n ϕ = { V x,r }r {0,1} c log n is a polynomial-sized qcsp instance. Since V runs in polynomial time, x ϕ is computable in polynomial time. By the completeness, if x 3SAT, then val(ϕ) = 1. By the soundness, if x / 3SAT, then val(ϕ) 1/2.

24 GAP = PCP Assume ρ-gapqcsp is NP-hard for some ρ < 1 and q N. Then this easily translates into a PCP with q queries, ρ soundness, and logarithmic randomness for any language L NP: Given an input x, the verifier V will run the reduction f (x) to obtain a qcsp instance ϕ = { ϕ i }i [m]. V expects π to be an assignment to the variables of ϕ, which it will verify by choosing a random i [m] and checking that ϕ i is satisfied (by making q queries). If x L, then the verifier will accept with probability 1. If x / L, it will accept with probability at most ρ.

25 NP PCP(poly(n), 1)

26 Walsh-Hadamard code The Walsh-Hadamard code encode bit strings of length n by linear functions in n variables over GF(2). Let u {0, 1} n. Then WH(u) = x u x = i [n] u i x i (mod2).

27 Random Subsum Principle Theorem Let u v. Then Pr x [ u x v x ] = 1/2.

28 Local testing of Walsh-Hadamard code Definition Let ρ R with 0 ρ 1. Two functions f, g : {0, 1} n {0, 1} are ρ-close if Pr x [ f (x) = g(x) ] ρ. f is ρ-close to a linear function if there exists a linear function g such that f and g are ρ-close. Theorem Let f : {0, 1} n {0, 1} be such that [ ] f (x + y) = f (x) + f (y) ρ Pr x,y for some ρ > 1/2. Then f is ρ-close to a linear function.

29 Local decoding of Walsh-Hadamard code Let δ < 1/4 and f : {0, 1} n {0, 1} be (1 δ)-close to some linear function f. Then f is uniquely determined. Suppose we are given x {0, 1} n and random access to f. Can we obtain the value f (x) using only a constant number of queries? 1. Choose x R {0, 1} n. 2. x x + x. 3. y f (x ) and y f (x ). 4. Output y + y. Lemma Pr [ f (x) = y + y ] 1 2δ.

30 Systems of quadratic equations over GF(2) (QUADEQ) Example The all one assignment satisfies u 1u 2 + u 3u 4 + u 1u 5 = 1 u 2u 3 + u 1u 4 = 0 u 1u 4 + u 3u 5 + u 3u 4 = 1. QUADEQ is the problem, given m n 2 matrix A and an m-dimensional vector b, of finding an n 2 -dimensional vector U satisfying 1. AU = b, 2. U is the tensor product u u of some n-dimensional vector u Lemma QUADEQ is NP-complete.