Solving BCSP using GA and PBIL Constraint Satisfaction Problem Group Sana Benhamida Andrea Roli Belgasem Ali Problem leader J.V.Hemert Jorge Tavares Group Leader Michele Sebag Samer Saadah Summer School

Evolutionary Optimization 1. Free Optimization 2.Constrained Optimization 3.Constraint Satisfaction

What is a constraint satisfaction problem? A Constraint Satisfaction Problem ( CSP) is a triplet where Z is a set of variables, D is a function that maps a finite set of objects of arbitrary type to Z. and C is a set of constraints that restrict certain simultaneous object assignments.

Objectives  Assign to each an object from such that no is violated  Find all possible instantiations of variables that do not violate a constraint  Prove that there is no solution for a given problem  Find a partial solution (as few violated constraints as possible) for an unsolvable problem.

EXAMPLES  Graph colouring: given a graph, find a k- colouring of the nodes such that nodes connected are colored with different colors.  n-Queens: given a n × n chess board and n queens, place the queens on the board such that no queen attacks another queen.  SAT: given a Boolean formula, find an assignment of variables such that the formula evaluates to true.

Example: graph-k colouring with k =3

Binary Constraint Satisfaction Problems Binary Constraint Satisfaction Problem (BINCSP) is a CSP where all constraints are associated with at most two variables. Given,

The problems  Generate random Binary CSP problem instances  With various levels of difficulty, controlled from: 1. Number of variables (n) 2. Domain size of each variable ( | D | = m) 3. Constraint Density (p1 or d) in [0,1] 4. Average tightness of a constraint (p2 or t), in[0,1]

Solving csps

Approaches  Simple GA scheme 1- Two-point crossover 2- Global crossover 3- Distribution based crossover  Population-Based Incremental Learning (PBIL, Baluja 1995) scheme.

Representation Problems Solutions Chromosome representation Domain size: between 2 and 15

Global Crossover (Gcx) 1 2 ……..……………………….7 a a d ……………

Distribution Based Crossover (DB-cx) x1 x2 x3 x4 x5 x6 x A B C A B C B C A C

1-Initialize probability matrix 2-Generate population 3- Evaluate population 4- Update probability matrix 5- Mutate probability matrix 6- Go to 2. PBIL SCHEME

Updating PBIL distribution Increase the probability of “good” assignments. Multiplicative updating rule with Learning Rate.

Parameters Population size: 10 Mutation probability: 0.05 Number of evaluations: 100,000 Number of runs for each instance :10 Number of instances: 25 for each value of t. Mutation probability for PBIL :

Results T/ AlGGA-2cxGA-GcxGA-DcxPBILSAW

Conclusion  Applying PBIL for Constraint Satisfaction  Promising Results for a set of BCSP instances.  GA approach is limited.

Future Research  More experiments with PBIL.  Better mutation operator (on the distribution)

Solving BCSP using GA and PBIL Constraint Satisfaction Problem Group Sana Benhamida Andrea Roli Belgasem Ali Problem leader J.V.Hemert Jorge Tavares Group Leader Michele Sebag Samer Saadah Summer School