1. Generating Hard Satisfiability Problems1 Bart Selman, David Mitchell, Hector J. Levesque Presented by Xiaoxin Yin

2. Generating Hard Satisfiability Problems 2 Can SAT be solved in polynomial time in average case? In Goldberg’s paper [Goldberg ‘79] it is claimed that SAT can be solved “on average” in polynomial time. Goldberg’s model of generating formulas  m clauses, n variables, each literal has probability p to be in each clause Is SAT really easy?

3. Generating Hard Satisfiability Problems 3 Can SAT be solved in polynomial time in average case? Goldberg’s SAT formulas are easy to solve [Franco & Paull ‘83] Theorem 1: Number of truth assignments for a formula is greater than 2 n (1-ε) with probability 1 Theorem 2: By randomly guessing truth assignments w times, Pr(success)=1 – m/(2 αnw ) assume each clause has at least αr literals

4. Generating Hard Satisfiability Problems 4 K-SAT is a harder problem K-SAT: each clause has K literals

5. Generating Hard Satisfiability Problems 5 Problem Definition of K-SAT Generate formulas of  N variables  K literals per clause (K = 3 in this paper)  M clauses To generate a clause  Randomly choose K distinct variables  Negate each with probability 0.5 Generate formulas with certain ratio of clauses-to-variables

6. Generating Hard Satisfiability Problems 6 DP Procedure DP: a backtracking depth-first search in the space of all truth assignments Procedure DP  Given a set of clauses Σ defined over a set of variables V  Set the value of a variable v and call DP on the simplified formula If this call returns “satisfiable”, then return “satisfiable”  Set v to the opposite value, return the result of calling DP on the re-simplified formula

7. Generating Hard Satisfiability Problems 7 Three Common Rules for DP The unit clause rule  If a clause contains only one literal, set it to true The pure literal rule  If a formula contains a literal but not its complement, set it to true The smallest clause rule  If none of the above rules applies, set a variable in a smallest clause The last two rules are not used in this paper

8. Generating Hard Satisfiability Problems 8 DP’s Performances on K-SAT Ratio of clauses-to-variables significantly affects the hardness of formulas

9. Generating Hard Satisfiability Problems 9 Ratio of Clauses-to-variables vs. Computational Cost Another example in [Mitchell & Levesque ‘96]

10. Generating Hard Satisfiability Problems 10 50% Point 50% point: given a certain N (# variables), the point that 50% of generated formulas are satisfiable 50% point is stable w.r.t. ratio of clauses-to- variables N2050100150 50% point4.554.364.314.3

11. Generating Hard Satisfiability Problems 11 50% Point – The Hardest Point (cont.) 50% point is close to the location of peak hardness

12. Generating Hard Satisfiability Problems 12 Satisfiable vs. Unsatisfiable Formulas Short formulas – under-constrained, have many satisfying assignments Long formulas – over-constrained, contradictions can often be easily found

13. Generating Hard Satisfiability Problems 13 Finding All Satisfying Assignments or Contradiction Let DP search the full space, until finding a contradiction Given a set of variables  Ratio of clauses to variables increases →Search space decreases

14. Generating Hard Satisfiability Problems 14 Finding All Satisfying Assignments or Contradiction (cont.) Computational cost of DP – monotonically decreases for increasing ratios of clauses to variables

15. Generating Hard Satisfiability Problems 15 Satisfiablity when ratio of clause-to- variable is small Pure literal rule: if a formula contains a literal but not its complement  Set this literal to 1  Remove all clauses containing this literal For 3-CNF with up to 1.63n clauses, pure literal rule by itself finds satisfying assignments with high probability [Broder, Frieze & Upfal ‘93]

16. Generating Hard Satisfiability Problems 16 Satisfiablity when ratio of clause-to- variable is small Smallest clause rule:  Choose a (random) literal in a (random) smallest clause For 3-CNF with less than 3.003n clauses, by smallest clause rule one can find satisfying assignments with high probability [Frieze & Suen ‘92] By “high probability” we mean Pr → 1 as n → ∞

17. Generating Hard Satisfiability Problems 17 Satisfiablity when ratio of clause-to- variable is large When c>4.762, a random 3-SAT formula is unsatisfiable with high probability [Kamath et al ‘94]. Some intuitions of proof:  Consider a certain assignment Z 1, each clause is true with probability 7/8.  Let #F denote number of satisfying assignments on F. E[#F] = 2 n (7/8) cn.  By Markov inequality, P[#F>0]≤E[#F]=(2∙(7/8) c ) n  This probability is exponentially small when c > 5.191

18. Generating Hard Satisfiability Problems 18 DP vs. Resolution DP searches for satisfying assignments as well as contradictions Resolution searches for contradictions Some result about resolution  In k-SAT problems of cn clauses, when k ≥ 3 and c2 –k ≥ 0.7, with probability tending to 1 as n goes to infinity, a randomly chosen formula of cn clauses is unsatisfiable, but there exists ε>0 such that every resolution proof must generate at least (1+ε) n clauses [Chvatal & Reed ‘92]  c=5.6 when k=3

19. Generating Hard Satisfiability Problems 19 Satisfiability for Different K For a random formula F k (n, cn)  c k = sup{ c : F k (n, cn) is satisfiable with high prob}  c k * = inf{ c : F k (n, cn) is unsatisfiable with high prob} [Kirousis et al ‘98] c k * ≤ 2 k ln2 – (1+ln2)/2 [Achlioptas and Peres ‘03] c k = c k * (1 – o(1)) c k ≥ 2 k ln2 – (k+1)ln2/2 – 1 – δ k (for a certain sequence δ k →0) ckck ck*ck* satisfiableunsatisfiableundetermined

20. Generating Hard Satisfiability Problems 20 Satisfiability for Different K (cont.) Lower and upper bounds for different K

21. Generating Hard Satisfiability Problems 21 K-SAT with different K’s When K is larger → Higher satisfiability, larger search space when n is large

22. Generating Hard Satisfiability Problems 22 P-SAT with different K’s Each variable has certain probability to appear in each clause Each clause has K variables on average

23. Generating Hard Satisfiability Problems 23 Solving SAT by Local Search [B. Selman et al ‘92] “A new method for solving hard satisfiability problems” GSAT – greedily search in the space of assignments GSAT algorithm Repeat for MAX-TRIES times randomly generate an assignment repeat for MAX-FLIPS times flip a variable to get largest increase in number of satisfied clauses

24. Generating Hard Satisfiability Problems 24 Solving SAT by Local Search (cont.) Performance of GSAT Drawback – cannot prove unsatisfiability

25. Generating Hard Satisfiability Problems 25 Thank you!

26. Generating Hard Satisfiability Problems 26 Additional Contents Performances of other approaches for solving K-SAT are presented in [Larrabee & Tsuji ‘93], which shows similar results (easy-hard- easy pattern)

27. Generating Hard Satisfiability Problems 27 Solving 2-SAT in Linear Time Choose a variable x and assign a value (e.g. x=1)  Remove all clauses that are true  Set values to all variables whose values are decided  Propagate in this way until nothing can be done  If contradiction happens, return false A set of clauses are left that are independent with the removed ones  If these clauses are satisfiable, return true  Else return false