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Complexity of Approximation

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della
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Complexity of Approximation. PCP Theory and Its Relation to Approximability. PCP (Probabilistically Checkable Proofs). An Extension of NP which has poly-time checkable proofs. A problem  is in NP if its membership can be proved as follows: -- The prover sends a proof of size p(n) to

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Complexity of Approximation
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  1. Complexity of Approximation PCP Theory and Its Relation to Approximability

  2. PCP (Probabilistically Checkable Proofs) • An Extension of NPwhich has poly-time checkable proofs. • A problem is in NP if its membership can be proved as follows: -- The prover sends a proof of size p(n) to the verifier. -- The verifier determines in poly-time whether the proof is valid. -- An instance x is in if and only if the prover can present a valid proof.

  3. A problem  is in PCP(f(n), g(n)) if a prover can proof its membership as follows: -- Prover sends p(n) bits of proofs to verifier. -- Verifier select randomly g(n) bits of the proof, and decides whether the input x is in  -- Verifier does this in poly-time, and uses only f(n) random bits. -- If x is in then the prover can always convince the verifier (i.e. success probab. = 1). -- If x not in then the success probab < ¼.

  4. Related Interactive Proof Systems IPS for Sat Input: Boolean formula Prover: send an assignment s of variables Verifier: verify that s satisfies F Error probability = 0 Random bits used = 0 t

  5. IPS for Graph Coloring Input: graph G, integer k > 0 Prover: send a mapping c from V1 to {1, 2, …, k} Verifier: select one edge {u,v}, check c(u) not= c(v) The chance of prover cheating and verifier catches it is 1/m. Repeat it for tm times, the chance of prover cheating And verifier not catches it is about 1/e t

  6. IPS for Graph Non-isomorphism Input: two graphs G1 and G2 Verifier: Select a random bit b, and a random permutation  on Vb; send (Gb) to prover Prover: send a bit b' to Verifier Verifier: accept if b = b' Error Probability: If G1 not iso G2, then prover Can always get b' = b if it is allowed full power Otherwise, prover has ½ chance to get b'=b.

  7. PCP Theorem NP = PCP( O(log n), O(1)). 3SAT in PCP( O(log n), O(1)) means there is a poly-time reduction that maps a 3-CNF formula F into another Boolean formula F' with the following property: F' = C1 C2 … Cm, and each Ci is a formula of k variables F is satisfiable There is an assignment that satisfies all Ci's in F' F is not satisfiable No assignment can satisfy more than ¼ of Ci's in F'

  8. Proof of PCP Theorem Proof 1. Based on the idea of interactive proof systems Main math tool: coding theory Proof 2. Based on the construction of expanders Main math tool: combinatorics

  9. Corollary of PCP Theorem NP = PCP( O(log n), O(1)). Max-3Sat has an NP-hard gap [1, r] for some r > 1.

  10. Hastad 3-Bit PCP Theorem • Max-3Sat is in PCP(O(log n), 3) --- the verifier only needs to check 3 bits of the proof given by the prover • Very useful for getting tight NP-hard gaps. • E.g., Max-3Lin has NP-hard gap [0.5+, 1-].

  11. Further applications of PCP Theorem Theorem Theorem Proved using PCP system

  12. Min-CDS Theorem

  13. Reduction from SC

  14. S has a sc of size k G has a cds of size k + 1 Suppose G has a (clog n)-approx alg. We can find a cds for G of size c log(m+n+2) (opt(S) + 1) We get a sc for S of size < c d (log n) opt(S) for sufficiently large n (and m < n)

  15. CLIQUE Theorem Proved using the proof of thePCP theorem.

  16. Other hard-to-approximate problems are all n -inapproximable 1-

  17. Unique Game Conjecture • Unique Game: Given a graph G with the vertex-coloring constraints: each vertex u having a function f: C  C (if u is red then its neighbors must be blue, etc), also given a number with the property that the coloring constraints can either be satisfied for  or for 1-portion of vertices (called a promise problem), determine which case is true. • Unique game conjecture states that this problem is NP-hard.

  18. Equivalent Formulation • A unique game is a problem in PCP(O(log n), 2) such that for the two queries asked by the verifier, the 2nd query (which may have k possible answers) has always a unique answer to make the verifier to accept. • The Unique Game Conjecture states that every problem in NP has a unique game.

  19. Applications of Unique Game Conjecture If the conjecture is true, then Min-VC does not have any r-approx for r < 2. current best bound: 1.36 Max-Cut does not have any r-approx for r < 1/0.87 current best bound: 1/0.94 …(many strange implications)