1 / 20

A non trivial (1- )-approximation alg. For Max-Sat

A non trivial (1- )-approximation alg. For Max-Sat. Ilan Newman Haifa University Joint work with: Guy Wolfovitz. Max-Sat: A CNF forula on n variables: F= AND(C 1 ,…, C m ) where each clause: C i = OR(t i1 ,t i2 ,…) and each t ij is a literal.

kamin
Download Presentation

A non trivial (1- )-approximation alg. For Max-Sat

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. A non trivial (1-)-approximation alg. For Max-Sat Ilan Newman Haifa University Joint work with: Guy Wolfovitz

  2. Max-Sat: A CNF forula on n variables: F= AND(C1,…, Cm) where each clause: Ci = OR(ti1,ti2,…) and each tijis a literal. Goal: Find an assignment to the variables that statisfies as many clauses as possible.

  3. If every clause is of length at most k, F is called k-CNF. Note: we don’t have any restriction on the clause length.

  4. Non-trivial Algorithms An Algorithm is non-trivial if there is a constant c such that the algorithm runs in (randomized) time at most cn for c< 2. (n is always the number of variables).

  5. Previous Results The trivial algorithm – time 2^n poly(m). NP-hard to decide satisfiability NP-hard to approximate 3-sat better than 7/8-factor (Håstad 1997).

  6. Many results on non-trivial algorithms for exact sat for O(1)-CNF formulae. [PPSZ05, CIKP03, IP01, IT03, D**02, HSSO-02, S99, BR99 …….]. Hirsch 2000 – a non trivial algorithm for (1-)-approximation for k-cnf in time cn for some c=c(k,) < 2. What about general CNF ?

  7. Results here An non-trivial algorithm for (1-)-approximation of max-sat in general CNF formulae. Inspiration: Hirsch algorithm that is based on Schöning algorithm.

  8. Schöning Algorithm – Alg. A • Pick a random assignment α1 to the variables. • Repeat for r=3n times: • choose a clause C that is not satisfied by αi. • choose a random literal t  C and flip its value in αito result inαi+1. • Return the αs that satisfies max no. of clauses.

  9. Pick a random assignment α1 to the variables • Repeat for r=3n times: • choose a clause C that is not satisfied by αi. • choose a random literal • t  C and flip its value in αi to result in αi+1. • Return the αr that satisfies max no. of clauses. Thm [Schöning] If F is k-CNF then, Prob(A finds a satisfying α) > [2(1-1/k)]-n Proof: Fix a satisfying assignment α* of F. We consider the hamming distance between αi and α* as a random walk on the line: S1,… ,Sr are the distances.

  10. Pick a random assignment α1 to the variables. • Repeat for r=3n times: • choose a clause C that is not satisfied by αi. • choose a random literal t  C and flip its value in αi to result in αi+1. • Return the αr that satisfies max no. of clauses. • Pr(Si = Si-1 +1) ≤ 1-1/k • Pr(Si = Si-1 -1) ≥ 1/k, • Pr(S1 = j) =( ) 2-n • Pr(success| S1=j) > (1/(k-1))j. • Pr(success) > (2(1-1/k))-n

  11. Hirsch Algorithm – max_sat *Input: F is k-CNF. • Pick a random assignment α to the variables. • Repeat for r=3n times: • choose a random clause C that is not satisfied by α. • choose a random literal t  C and flip its value in α. • Return the αi that satisfies max no. of caluses.

  12. Analysis: Let Br = the event that αr satisfies less than (1- )opt(F) clauses. Let B =  Br , let α* be the assignment than achieves opt(F). As long as B occurs the random walk behaves exactly as for k-CNF with prob of going left = /(2k).

  13. This implies, Pr( ( i, αi = α*) U BC) > (2- /2k)-n.

  14. Our Result Def. – A clause C is δ-critical w.r.t a (partial) assignment α if α satidfies at most δ-fraction of the literals in C.

  15. Alg.F is a general CNF formula. • F1=F, n1=n, • For i=1 to δn • vi = max_sat(Fi) + |F|-|Fi|+ ξi • Let Ci = set of caluses of Fi of length at least 1/ δ. • If Ci Ø choose a random C  Ci and a random δ-critical assignment α for C. • Let Fi+1 be Fi(α). • Return max vi

  16. F1=F, n1=n, • For i=1 to δn • vi = max_sat(Fi) + |F|-|Fi| +ξi • Let Ci = set of caluses of Fi of length at least 1/ δ. • If Ci Ø choose a random • C  Ci and a random δ-critical assignment α for C. • Let Fi+1 be Fi(α). • Return max vi Analysis: Let α*be the opt-assignment. A success will be that some vi satisfies at least (1- )opt(F) , or that we have assigned correctly (as inα*) k variables in a clause (w. prob. 2-ck, for c<1). Let Ai = “vi > (1- )opt(F) “ Let Si = U{k ≥i} Ak. Pr(suc.) > min( Pr(A1| B1), Pr(S2|B1C)) Let Bi = “at most μ-fraction of the clauses in Ci are δ-critical w.r.t. α*“.

  17. Bounding Pr(A1| B1): If B1 occurs then the probability of choosing a non-critical clause in max_sat, and choosing the `right’ literal is f(, μ)). Hence, going left in the random walk is at least (2- f(, μ))-n.

  18. Bounding Pr(S2|B1C) with respect to F2 Since B1C occurred, the clause chosen at step 5 is δ-critical w.r.t. α* with probability at least μ. If this happens, then α chosen at step 6 is δ-critical assignment for C, with probability 2-H(δ)k and we delete k variables for F – -> F2. • F1=F, n1=n, • For i=1 to δn • vi = max_sat(Fi) + |F|-|Fi| • Let Ci = set of caluses of Fi of length at least 1/ δ. • If Ci Ø choose a random • C  Ci and a random δ-critical assignment α for C. • Let Fi+1 be Fi(α). • Return max vi

  19. Pr(S2)> min(Pr(A2| B2,), Pr(S3|B2C,)) We now bound Pr(A2| B2) with respect toF2assuming thatF2is already restricted to the correct kvariables in the same way….QED

  20. Open Problems • Find a non-trivial algorithm for general CNF. • Find a better approximation alg. --- can one use the structure of the formula.

More Related