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

Stocs – A Stochastic CSP SolverStocs – A Stochastic CSP Solver

Bella Dubrov

IBM Haifa Research Lab

© Copyright IBM


Copyright IBM2

Outline

CSP solving algorithms Systematic Stochastic

Limitations of systematic methods Stochastic approach Stocs algorithm Stocs challenges Summary


Copyright IBM3

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


Copyright IBM4

CSP Solving algorithms

Systematic: GEC

Stochastic: Stocs


Copyright IBM5

Systematic approach

Systematically go over the search space Use pruning whenever possible

pruning is done by projection


Copyright IBM6

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}


Copyright IBM7

Limitations of systematic methods: example 1

cba

Ncba

*

1,...,0,,

Propagation is hard (factoring)


Copyright IBM8


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)


Copyright IBM9


10.4

00.3

00.2

00.1

ac

cb

ca

ba

0,1 cbaOnly solution:

1,...,0,, Ncba

Local consistency at onset probability of success: 1/N


Copyright IBM10

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


Copyright IBM11

Stochastic approach

General idea: Start from some stateFind the next state and move thereStop 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 annealingTabu search


Copyright IBM12

Stocs algorithm overview

Check states on length-scales “typical” for the problem. Hop to a new state if cost is lowerLearn the topography of the problem: learn the typical step sizes and directionsGet domain-knowledge as input strategies


Copyright IBM13

Problem:7 groups of players 6 members in each groupPlay 4 weeksWithout any two players playing together (in the same group) twice

Exponential decreaseNo sense in trying step sizes larger than 20. But may benefit strongly from step sizes of 10-15

Reproducible - characterizes the problem

Social Golfer Problem (7-6-4)

1

10

100

1000

10000

0 10 20 30 40 50

Step Size

Nu

mb

er o

f Su

cces

ses

4.6 million tries 4.7 million tries 93 million tries

Example: Social Golfer Problem


Copyright IBM14

Problem:Minimize the autocorrelation on a sequence of N (45) bits

Non-exponential decrease, followed by saturationMakes sense to always try large steps

Identifies small characteristic features

Extremely reproducible

Example: LABS

Low Autocorrelation Binary Squence (N = 45)

1000

10000

100000

0 10 20 30 40 50

Step Size

Nu

mb

er o

f Su

cces

ses

77 million tries 77 million tries


Copyright IBM15

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

Selection Problem (40,30)

1

10

100

1000

10000

0 500 1000 1500

Step Size

Num

ber o

f Su

cces

ses

1.5 million tries 1.5 million tries 3.3 million tries


Copyright IBM16

Problem:Same as before, modeled differently

Prefer intermediate step sizes Reproducible

Example: Selection Problem, different modeling

Selection Problem (4,3)

10000

100000

1000000

10000000

0 2 4 6 8 10 12 14

Step Size

Nu

mb

er o

f S

ucc

esse

s

n tries n tries m tries


Copyright IBM17

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


Copyright IBM18

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


Copyright IBM19

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


Copyright IBM20

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


Copyright IBM21

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


Copyright IBM22

Thank you

Documents

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