1. AAAI00 Austin, Texas Generating Satisfiable Problem Instances Dimitris Achlioptas Microsoft Carla P. Gomes Cornell University Henry Kautz University of Washington Bart Selman Cornell University

2. Introduction An important factor in the development of search methods is the availability of good benchmarks. Sources for benchmarks: Real world instances –hard to find –too specific Random generators –easier to control (size/hardness)

3. Random Generators of Instances Understanding threshhold phenomena lets us tune the hardness of problem instances: At low ratios of constraints - most satisfiable, easy to find assignments; At high ratios of constraints - most unsatisfiable easy to show inconsistency; At the phase transition between these two regions roughly half of the instances are satisfiable and we find a concentration of computationally hard instances.

4. Limitation of Random Generators PROBLEM: evaluating incomplete local search algorithms Filtering out Unsat Instances - use a complete method and throw away unsat instances. Problem: want to test on instances too large for any complete method! “Forced” Formulas Problem: the resulting instances are easy – have many satisfying assignments

5. Outline I Generation of only satisfiable instances II New phase transition in the space of satisfiable instances III Connection between hardness of satisfiable instances and new phase transition IV Conclusions

6. Generation of only satisfiable instances

7. Given an N X N matrix, and given N colors, color the matrix in such a way that: -all cells are colored; - each color occurs exactly once in each row; - each color occurs exactly once in each column; Quasigroup or Latin Square Quasigroup or Latin Squares

8. Quasigroup Completion Problem (QCP) Given a partial assignment of colors (10 colors in this case), can the partial quasigroup (latin square) be completed so we obtain a full quasigroup? Example: 32% preassignment

9. QCP: A Framework for Studying Search NP-Complete. Random instances have structure not found in random k-SAT Closer to “real world” problems! Can control hardness via % preassignment BUT problem of creating large, guaranteed satisfiable instances remains… (Anderson 85, Colbourn 83, 84, Denes & Keedwell 94, Fujita et al. 93, Gent et al. 99, Gomes & Selman 97, Gomes et al. 98, Shaw et al. 98, Walsh 99 )

10. Quasigroup with Holes (QWH) Given a full quasigroup, “punch” holes into it Difficulty: how to generate the full quasigroup, uniformly. 32% holes Question: does this give challenging instances?

11. Markov Chain Monte Carlo (MCMM) We use a Markov chain Monte Carlo method (MCMM) whose stationary (egodic) distribution is uniform over the space of NxN quasigroups (Jacobson and Matthews 96). Start with arbitrary Latin Square Random walk on a sequence of Squares obtained via local modifications

12. Generation of Quasigroup with Holes (QWH) 1)Use MCMM to generate solved Latin Square 2)Punch holes - i.e., uncolor a fraction of the entries The resulting instances are guaranteed satisfiable QWH is NP-Hard Is there % holes where instances truly hard on average?

13. Easy-Hard-Easy Pattern in Backtracking Search % holes Computational Cost Complete (Satz) Search Order 30, 33, 36 QWH peaks near 32% (QCP peaks near 42%)

14. Easy-Hard-Easy Pattern in Local Search % holes Computational Cost Local (Walksat) Search Order 30, 33, 36 First solid statistics for overconstrainted area!

15. Phase Transition in QWH? QWH - all instances are satisfiable - does it still make sense to talk about a phase transition? The standard phase transition corresponds to the area with 50% SAT/UNSAT instances Here all instances SAT Does some other property of the wffs show an abrupt change around “hard” region?

16. Backbone Preassigned cells Number sols = 4 Backbone Backbone is the shared structure of all solutions to a given instance (not counting preassigned cells) Backbone size = 2

17. Phase Transition in the Backbone We have observed a transition in the size of backbone Many holes – backbone close to 0% Fewer holes – backbone close to 100% Abrupt transition – coincides with hardest instances!

18. New Phase Transition in Backbone % Backbone Sudden phase Transition in Backbone and it coincides with the hardest area % holes Computational cost % of Backbone

19. Why correlation between backbone and problem hardness? Intuitions: Local Search Near 0% Backbone = many solutions = easy to find by chance Near 100% Backbone = solutions tightly clustered = all the constraints “vote” in same direction 50% Backbone = solutions in different clusters = different clauses push search toward different clusters (Current work – verify intuitions!)

20. Why correlation between backbone and problem hardness? Intuitions: Backtracking search Bad assignments to backbone variables near root of search tree cause the algorithm to deteriorate For the algorithm to have a significant chance of making bad choices, a non-negligible fraction of variables must appear in the backbone

21. Reparameterization of Backbone % of Backbone Backbone for different orders (30 - 57)

22. Reparameterization Computational Cost Computational Cost different orders (30, 33, 36) % of Backbone Local Search (normalized) Local Search (normalized & reparameterized)

23. Summary QWH is a problem generator for satisfiable instances (only): Easy to tune hardness Exhibits more realistic structure Well-suited for the study of incomplete search methods (as well as complete) Confirmation of easy-hard-easy pattern in computational cost for local search New kind of phase transition in backbone Reparameterization GOAL – new insights into practical complexity of problem solving

24. QWH generator, demos, available soon (< one month): www.cs.cornell.edu/gomes www.cs.washington.edu/home/kautz SATLIB CSPLIB

25. Parameterization % of Backbone Backbone for different orders (30 - 57)