1. Relaxed DPLL Search for MaxSAT (short paper) Lukas Kroc, Ashish Sabharwal, Bart Selman Cornell University SAT-09 Conference Swansea, U.K. July 3, 2009

2. SAT and MaxSAT  SAT: Given a Boolean formula such as find, if possible, a truth assignment such that F is satisfied.  MaxSAT: Given a Boolean formula, find a truth assignment that satisfies as many constraints as possible  Many applications in combinatorial and probabilistic reasoning (e.g., critical failure paths; as a tool within Markov Logic Networks)  Traditional approaches for MaxSAT:  Systematic search: complete but limited scalability (e.g., MiniMaxSat [Heras et al. ’07], Msuf [Marques-Silva et al.’08], MaxSatz [Li et al.’07] )  Local search: scalable but sometimes far from optimum (e.g., Walksat [Selman-Kautz-Cohen ’96], GLS [Mills et al. ’00], SAPS [Hutter-Tompkins-Hoos ’02- 03], Adaptg2wsat+p [Li-Wei-Zhang ’07] ) Ashish Sabharwal2 x y SatCls 0 03 0 14 1 04 1 15 SAT 2009

3. Local Search: “Natural” for MaxSAT?  Local search methods walk on a landscape composed of truth assignments  Height = number of unsatisfied constraints  At every point in time, have a (sub-optimal) MaxSAT solution!  “anytime solution” to MaxSAT  Bottleneck: local minima, just as in local search for SAT SAT 2009Ashish Sabharwal3 [credit: reactive-search.org]

4. Systematic Search: 2 Key Features The DPLL process, and extensions, for SAT  Search guided by  Heuristics  Clauses falsified or made “critical” during search  Efficiency relies heavily on the ability to avoid local inconsistencies Unit propagation: if a=0, b=0, and have clause (a or b or  c), better force c=1 right away Clause learning: if a search branch reaches a contradiction, learn why this happened and never let this happen again SAT 2009Ashish Sabharwal4

5. Systematic Search: 2 Key Features Unfortunately, unit propagation and clause learning are not suitable for DPLL-based MaxSAT solvers  For MaxSAT, it may be best to ignore the current local inconsistency and try to satisfy the rest of the formula as much as you can  Systematic DPLL-style MaxSAT solvers necessarily do not employ traditional unit propagation or clause learning [note: may use more sophisticated resolution-based inference] Cost? Systematic MaxSAT solvers are significantly less scalable than systematic SAT solvers e.g., verification instance cmu-bmc-barrel6.cnf : – best MaxSAT solvers need 20-30 minutes – MiniSat needs ~2 seconds SAT 2009Ashish Sabharwal5

6. The Question(s) SAT 2009Ashish Sabharwal6 Can state-of-the-art systematic SAT techniques be used as (incomplete) effective heuristics for MaxSAT? Aside: What do “best possible” near-solutions of unsatisfiable industrial SAT instances (~100K variables, 100K-1M clauses) look like? 1000’s of unsatisfied clauses or just a few? 1.Perhaps as a guidance mechanism for local search? 2.To zoom down to near-satisfying assignments? Yes! [upcoming paper at IJCAI-09] Effort driven by local search, guided by DPLL running in parallel Yes! [this talk] Effort driven by “relaxed” DPLL search, followed by local search if needed

7. Proposed Solver: RelaxedMinisat + Walksat Context:  Suited to MaxSAT instances non-trivial to prove unsatisfiable (otherwise method falls back to pure local search)  Instances with “near-solutions” (≤ 200 or so violated clauses) Main idea: 1.Use “relaxed” DPLL (w/ CL) as the main MaxSAT search engine  Rexation = tolerate “k” conflicts before backtracking 2.Report truth assignment with at most k conflicts on a branch 3.Improve it with local search, if needed “Theorem”: sound ( even with unit prop., clause learning, etc., solution found by RelaxedMinisat violates at most k clauses) but not complete SAT 2009Ashish Sabharwal7

8. Proposed Solver: RelaxedMinisat + Walksat Solver “instantiation”: ― Minisat for base DPLL search; Walksat for local search ― #conflicts to tolerate (small) fixed through quick binary search ― Rapid restarts helpful (every 100 backtracks) Key features:  Very easy to code: a few lines on top of Minisat; no change to Walksat  Running time very close to that of ‘pure’ Minisat  provides “bottleneck constraints” in the time comparable to prove unsatisfiability  Reveals a surprising feature of industrial unsat. instances not detected by any other current MaxSAT solver SAT 2009Ashish Sabharwal8

9. Experimental Evaluation

10. Results Summary  Simple idea; surprisingly good performance in practice (on instances non-trivial to prove unsatisfiable)  Tested on all 52 unsat. industrial instances from SAT Race-08 [more on this later]  Substantially better quality solutions, and faster: Often finds a solution with “very few” violated clauses within seconds – best systematic methods time out after one hour – best local search suggests 100’s to 1000’s of violated clauses SAT 2009Ashish Sabharwal10

11. Experimental Setup Comparison against:  Systematic solvers with best performance in MaxSAT Evaluation 2007 (maxsatz, msuf)  Local search solvers from UBCSAT with best performance on our suite (saps, adaptg2wsat+p)  Hybrid technique to be presented at IJCAI-09 (MiniWalk) [Time limit = 1 hour] SAT 2009Ashish Sabharwal11

12. Experimental Results Ashish Sabharwal12SAT 2009 Timelimit: 1 hour Surprising observation: Most unsat instances from SAT RACE ‘08 competition have assignments with only ONE unsatisfied clause!

13. Experimental Results: continued Ashish Sabharwal13SAT 2009 Timelimit: 1 hour

14. A Note on Benchmarks Industrial SAT Race instances clearly have interest, but… would have liked to compare on “standard” MaxSAT benchmarks from MaxSAT competitions, etc. “Unfortunately,” they are too easy for current SAT solvers (proved unsatisfiable in a couple of seconds)  RelaxedMinisat falls back to pure Walksat Is this really a true reflection of MaxSAT problem domains? Or a chicken-and-egg problem?  Perhaps there aren’t hard-to-prove-unsatisfiable MaxSAT benchmarks because no current MaxSAT solvers can handle them? SAT 2009Ashish Sabharwal14

15. A Note on Alternatives  Would more time have helped?  In retrospect, how about SAT re-encoding allowing one violated clause?  Create F’: satisfiable iff F has a near-solution violating 1 clause  Need one new variable per clause of F  100K+ additional vars  Need a way to count violated clauses naïve way: (100K+) 2 clauses cascading count: additional variables, limited propagation  doesn’t scale SAT 2009Ashish Sabharwal15 Local search appears to stabilize at highly sub-optimal value Not at all within the reach of systematic MaxSAT solvers

16. Summary and Future Directions  Simple, low-overhead strategy yields promising, and surprising, results!  DPLL-based systematic SAT solvers can provide effective guidance for MaxSAT search  Good at recognizing “near-solutions”, especially on interesting industrial instances  Creation of MaxSAT benchmarks that are hard to prove unsatisfiable  Do these single “bottleneck constraints” give any useful information?  Luckily, do not appear to be “goal constraints”; usually more interesting  Different bottleneck constraints for different random seeds  What does this say about our SAT encodings? Brittle encodings?  Dynamic parameter tuning and other optimizations [cf. Mate Soos for crypto-specific optimizations]  Weighted MaxSAT SAT 2009Ashish Sabharwal16