1. Algorithms for SAT Based on Search in Hamming Balls Author : Evgeny Dantsin, Edward A. Hirsch, and Alexander Wolpert Speaker : 羅正偉

2. Outline  Definition of SAT problem  Introduction of this paper  Definitions and notation  Randomized Algorithm

3. Preliminary: K-SAT problem

4. Preliminary-A fact  2-SAT is in P, but 3-SAT is NP-complete.

5. Preliminary-Hamming Ball  Definition: The Hamming ball of radius R around a truth assignment A is the set of all assignments whose Hamming distance to A is less than or equal to R.  The assignment A is called the center of the ball.  The volume of a ball is the number of assignments that belong to the ball.

6. Introduction  In this paper a randomized algorithm for SAT is given, and its derandomized version is the first non-trivial bound for a deterministic SAT algorithm with no restriction on clause length.  For k-SAT, Schuler’s algorithm is better than this one.

7. Notation

8. Fact about H(x)  The graph of H(x) is

9. A bound of V(n,R)

10. Randomized Algorithm  The randomized algorithm is called Random- Balls, and it invokes procedures called Ball- Checking and Full-Ball-Checking.

11. Ball-Checking algorithms

12. Observations  The recursion depth is at most R  Any literal is altered at most once during execution of Ball-Checking, and at any time, those remaining variables are assigned as in the original assignment

13. Lemma 1  There is a satisfying assignment in B(A,R) iff Ball-Checking(F,A,R) returns a satisfying assignment in B(A,R)  Proof. By induction of R (see reference)

14. Lemma2  The running time of Ball-Checking(F,A,R) is at most, where k is the maximum length of clauses occurring at step 3 in all recursive calls.  Proof. The recursion depth is at most R and the maximum degree of branching is at most k.

15. Full Ball Checking  Procedure Full-Ball-Checking(F,A,R) Input: formula F over variables, assignment A, number R Output: satisfying assignment or “no”  1.Try each assignment A’ in B(A,R), if it satisfies F, return it.  2.Return “no”

16. Observation  Full-Ball-Checking runs in time poly(n)mV(n,R)

17. Randomized algorithm

18. Lemma 3  Correctness of Random-Balls. For any R, l, the following holds: (a)If F is unsatisfiable, then Random-Balls returns “no”, (b)else Random-Balls finds a satisfying assignment with probabability 1/2

19. Proofs of lemma 3

20. A bound of V(n,R) REVIEW!!

21. Probability of correctness  So the prob. of correctness = 1 – 1/e > 1/2  Choosing N to be k times larger will reduce the probability of error to less than.  Note this doesn’t ruin the time complexity we need.

22. Lemma 4  Consider the execution of Random- Balls(F,R,l) that invokes Ball-Checking. For any input R, l, the maximum length of clauses chosen at step 3 of Procedure Ball- Checking is less than l.

23. Ball-Checking algorithms REVIEW!!

24. Proof of lemma 4

25. Lemma 5  For any R, l, let p be the probability (taken over random assignment A) that Random- Balls invokes Full-Ball-Checking. Then

26. Proof of lemma 5

27. Proof of lemma 5(conti.)

28. Fact about H(x)  The graph of H(x) is REVIEW!!

29. Theorem 1

30. Proof of theorem 1

31. Proof of Theorem 1(conti.)

32.  Assign R=a,l=b,where a < b constants. We use the fact ln(1+x)=x+o(x).

33. Proof of theorem1(conti.)

34. Proof of Theorem 1 (conti.)  Taking a=0.339, b=1.87, we have Φ,ψ>0.712, proving the theorem.

35. Reference  E. Dantsin, A. Goerdt, E. A. Hirsch, R. Kannan, J. Kleinberg, C. Papadimitriou, P. Raghavan, and U. Schoning. A deterministic algorithm for K-SAT based on local search. Theoretical Computer Science, 289(1):69-83, October 2002.