1. Carla P. Gomes CS4700 CS 4700: Foundations of Artificial Intelligence Carla P. Gomes gomes@cs.cornell.edu Module: Instance Hardness and Phase Transitions (Reading R&N: page 224-225)

2. Carla P. Gomes CS4700 Instance Hardness

3. Beyond NP-Completeness NP-Completeness is a worst-case notion! Not all problems instances are the same! We now have means for discriminating easy from hard instances  structural differences between instances of the same problem class.

4. Are all the Latin Square Instances (of same size) Equally Difficult? 1820150 Number Backtracks: 165 What is the fundamental difference between instances?

5. Are all Latin Square Instances Equally Difficult? 1820165 40% 50% 150 Number Backtracks: 35% Fraction of preassignment:

6. Complexity of Latin Square Completion Fraction of pre-assignment Median Runtime (log scale) Critically constrained area Overconstrained area Underconstrained area 42%50%20%

7. Phase Transition Almost all unsolvable area Fraction of pre-assignment Fraction of unsolvable cases Almost all solvable area Complexity Graph Phase transition from almost all solvable to almost all unsolvable

8. Carla P. Gomes CS4700 Latin Squares with Holes Given a full Latin Square, “punch” holes into it Difficulty: how to generate the full quasigroup, uniformly. 32% holes Question: does this give challenging instances?

9. Carla P. Gomes CS4700 Markov Chain Monte Carlo (MCMM) We use a Markov chain Monte Carlo method (MCMM) whose stationary (ergodic) distribution is uniform over the space of NxN Latin Squares (Jacobson and Matthews 96). –Start with arbitrary Latin Square –Random walk on a sequence of Squares obtained via local modifications

10. Carla P. Gomes CS4700 Generation of Latin Squares with Holes (LSWH) 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 –LSWH is NP-Hard Is there % holes where instances are truly hard on average?

11. Carla P. Gomes CS4700 Easy-Hard-Easy Pattern in Backtracking Search % holes Computational Cost Order 30, 33, 36 Peak near 32% (LSCP peaks near 42%) Research Question: why the peak?

12. Carla P. Gomes CS4700 Easy-Hard-Easy Pattern in Local Search % holes Computational Cost Local (Walksat) Search Order 30, 33, 36 First solid statistics for overconstrainted area! Research Question: why the peak?

13. Carla P. Gomes CS4700 These results for Latin Squares - a structured problem - nicely complement previous results on phase transition and computational complexity for random instances such as SAT, Graph Coloring, etc.

14. Carla P. Gomes CS4700 Propositional Satisfiability problem (SAT) Satifiability (SAT): Given a formula in propositional logic, is there a model (i.e., a satisfying interpretation, an assignment to its variables) making it true? We consider clausal form, e.g.: ( a   b   c ) AND ( b   c) AND ( a  c) possible assignments SAT: prototypical hard combinatorial search and reasoning problem. Problem is NP-Complete. (Cook 1971) Surprising “power” of SAT for encoding computational problems.

15. The phase transition Complexity peak Run time SAT phase UNSAT phase of interest for alg. design Prob. satisfiable Ratio of Clauses to Varables Early Results Thanks Bart Selman! 3 SAT (all clauses have 3 literals) More about Propositional Logic and SAT in the Logic module

16. Carla P. Gomes CS4700 Random 3-SAT as of 2005 Random Walk DP DP’ Walksat SP Linear time algs. GSAT Phase transition Mitchell, Selman, and Levesque ’92

17. Carla P. Gomes CS4700 Random 3-SAT as of 2005 Random Walk DP DP’ Walksat SP Linear time algs. GSAT Upper bounds by combinatorial arguments (’92 – ’05) 5.19 5.081 4.762 4.596 4.506 4.601 4.643