1. Stocs – A Stochastic CSP Solver Bella Dubrov IBM Haifa Research Lab © Copyright IBM

2. IBM Haifa Research Lab Copyright IBM 2 Outline  CSP solving algorithms  Systematic  Stochastic  Limitations of systematic methods  Stochastic approach  Stocs algorithm  Stocs challenges  Summary

3. IBM Haifa Research Lab Copyright IBM 3 Constraint satisfaction problems  Variables: Anna, Beth, Cory, Dave  Domains: Red, Green, Orange, Yellow houses  Constraints:  The Red and Green houses are in the city  The Orange and Yellow houses are in the countryside  The Red and Green houses are neighboring, as well as the Orange and Yellow houses  Anna and Dave have dogs, Beth owns a cat  Dogs and cats cannot be neighbors  Dogs must live in the countryside  Solution:  Anna lives in the Orange house, Beth lives in the Red house, Cory lives in the Green house, Dave lives in the Yellow house

4. IBM Haifa Research Lab Copyright IBM 4 CSP Solving algorithms Systematic: GEC Stochastic: Stocs

5. IBM Haifa Research Lab Copyright IBM 5 Systematic approach  Systematically go over the search space  Use pruning whenever possible  pruning is done by projection

6. IBM Haifa Research Lab Copyright IBM 6 Example: projector for multiply a x b = c a є [2, 20], b є [3, 20], c є [1, 20] Projection to input 1 a’ = [2, 6] Projection to input 2 b’ = [3, 10] Projection to result c’ = {6, 8, 9, 10, 12, 14, 15, 16, 18, 20}

7. IBM Haifa Research Lab Copyright IBM 7 Limitations of systematic methods: example 1 Propagation is hard (factoring)

8. IBM Haifa Research Lab Copyright IBM 8 Limitations of systematic methods: example 2  The Some-Different constraint: given a graph on the variables, the variables connected by an edge must have different values  Propagation is NP-hard for domains of size k > 3 (k-colorability)

9. IBM Haifa Research Lab Copyright IBM 9 Limitations of systematic methods: example 3 Only solution:  Local consistency at onset  probability of success: 1/N

10. IBM Haifa Research Lab Copyright IBM 10 Stochastic approach  State: an assignment of values to all the variables  Cost: a function from the set of states to {0} U R +  Cost = 0 iff all constraints are satisfied by the state

11. IBM Haifa Research Lab Copyright IBM 11 Stochastic approach  General idea:  Start from some state  Find the next state and move there  Stop if a state with cost 0 is found  Stochastic algorithms are usually incomplete  Different stochastic algorithms use different heuristics for finding the next state  Examples:  Simulated annealing  Tabu search

12. IBM Haifa Research Lab Copyright IBM 12 Stocs algorithm overview  Check states on length-scales “typical” for the problem. Hop to a new state if cost is lower  Learn the topography of the problem: learn the typical step sizes and directions  Get domain-knowledge as input strategies

13. IBM Haifa Research Lab Copyright IBM 13  Problem:  7 groups of players  6 members in each group  Play 4 weeks  Without any two players playing together (in the same group) twice  Exponential decrease  No sense in trying step sizes larger than 20.  But may benefit strongly from step sizes of 10-15  Reproducible - characterizes the problem Example: Social Golfer Problem

14. IBM Haifa Research Lab Copyright IBM 14  Problem:  Minimize the autocorrelation on a sequence of N (45) bits  Non-exponential decrease, followed by saturation  Makes sense to always try large steps  Identifies small characteristic features  Extremely reproducible Example: LABS

15. IBM Haifa Research Lab Copyright IBM 15  Problem:  Select different values for three variables out of a given set of values (smaller than domains)  Easy problem: results are for many runs  Prefer larger step sizes  (up to a cutoff)  Reproducible Example: Selection Problem

16. IBM Haifa Research Lab Copyright IBM 16  Problem:  Same as before, modeled differently  Prefer intermediate step sizes  Reproducible Example: Selection Problem, different modeling

17. IBM Haifa Research Lab Copyright IBM 17 Stocs algorithm At each step: decide attempt type: random, learned or user-defined if random: choose a random step if learned: decide learn-type: step-size, direction, … if step-size: choose a step-size which was previously successful (weighted) create a random attempt with chosen step size if direction: choose a direction which was previously successful (weighted) create a random attempt with chosen direction if user-defined: get next user-defined attempt

18. IBM Haifa Research Lab Copyright IBM 18 Optimization problems  Constraints must be satisfied  In addition, an objective function that should be optimized is given  Example: doll houses  Constraints as before  In addition each doll has a preferred set of houses  The best solution satisfies as much of the preferences as possible

19. IBM Haifa Research Lab Copyright IBM 19 Optimization with Stocs  Last year we added the optimization capability to Stocs  Optimization is natural for Stocs:  First find a solution  Then keep searching for a better state  Implementation:  Cost function from a state to a pair of non-negative numbers (c1, c2):  c1 is the cost of the constraints  c2 is the value of the objective function  lexicographic order on the pairs:  a better state will always improve the constraints  after a state with c1 = 0 is found, Stocs will continue searching for a better c2

20. IBM Haifa Research Lab Copyright IBM 20 Preprocessing and initialization  Before starting the search 2 things happen:  Preprocessing of the problem  including:  finding bits that should be constant in any solution  removing unnecessary variables  simplifying constraints  has a big impact on the search:  last year we improved the performance by a factor of 100 with preprocessing  Initialization: finding the initial state  Starting the search at a good state is critical  Currently, each constraint tries to initialize its variables to a satisfying assignment, considering the “wishes” of other constraints

21. IBM Haifa Research Lab Copyright IBM 21 Summary  Limitations of systematic methods  Stochastic approach: move between full assignments  Stocs: learn the topography of the problem, allow user-defined heuristics  Optimization with Stocs  Preprocessing and initialization  Variable types

22. IBM Haifa Research Lab Copyright IBM 22 Thank you