A Study into Ant Colony Optimisation, Evolutionary Computation and - - PowerPoint PPT Presentation

A Study into Ant Colony Optimisation, Evolutionary Computation and Constraint Programming on Binary Constraint Satisfaction Problems

Jano van Hemert Christine Solnon http://www.cwi.nl/∼jvhemert http://www710.univ-lyon1.fr/∼csolnon CWI, Amsterdam, The Netherlands LIRIS, University of Lyon 1, France

Jano van Hemert & Christine Solnon – p.1

Research questions

✔ For binary CSP, how do EC, ACO and CP relate to each

• ther performance wise?

Jano van Hemert & Christine Solnon – p.2

Research questions

✔ For binary CSP, how do EC, ACO and CP relate to each

• ther performance wise?

✔ ...near the phase transition

Jano van Hemert & Christine Solnon – p.2

Research questions

✔ For binary CSP, how do EC, ACO and CP relate to each

• ther performance wise?

✔ ...near the phase transition ✔ ...when the problem size is scaled up

Jano van Hemert & Christine Solnon – p.2

Research questions

✔ For binary CSP, how do EC, ACO and CP relate to each

• ther performance wise?

✔ ...near the phase transition ✔ ...when the problem size is scaled up ✔ What reasons lie behind failing?

Jano van Hemert & Christine Solnon – p.2

Binary constraint satisfaction & problem difficulty

Jano van Hemert & Christine Solnon – p.3

Constraint satisfaction

✔ Given a set of variables X, a set of finite domains D and a set of constraints C (Tsang, 1993)

Jano van Hemert & Christine Solnon – p.4

Constraint satisfaction

✔ Given a set of variables X, a set of finite domains D and a set of constraints C (Tsang, 1993) ✔ Each xi ∈ X has a corresponding discrete domain Dxi from which they can be assigned the value vi, denoted as the tuple xi, vi, where vi ∈ Dxi

Jano van Hemert & Christine Solnon – p.4

Constraint satisfaction

✔ Given a set of variables X, a set of finite domains D and a set of constraints C (Tsang, 1993) ✔ Each xi ∈ X has a corresponding discrete domain Dxi from which they can be assigned the value vi, denoted as the tuple xi, vi, where vi ∈ Dxi ✔ Every element c ∈ C is a constraint over a subset of variables of X, it consists of sets that contain tuples of values that are not allowed to be assigned simultaneously

Jano van Hemert & Christine Solnon – p.4

CSP: the problem

☞ Assign to each xi ∈ X a value from Dxi such that no constraint c ∈ C is violated

Jano van Hemert & Christine Solnon – p.5

CSP: the problem

☞ Assign to each xi ∈ X a value from Dxi such that no constraint c ∈ C is violated ✎ A solution is written as:

• x1, v1x2, v2 · · · x|X|, v|X|
• Jano van Hemert & Christine Solnon – p.5

CSP: the problem

☞ Assign to each xi ∈ X a value from Dxi such that no constraint c ∈ C is violated ✎ A solution is written as:

• x1, v1x2, v2 · · · x|X|, v|X|
• ☞ A binary constraint satisfaction problem is a CSP where

all constraints are associated with at most two variables

Jano van Hemert & Christine Solnon – p.5

Binary constraint satisfaction

☞ Using four parameters it is possible to make predictions about the difficulty of solving binary CSPs (Smith, 1993; Gent et al., 1996)

Jano van Hemert & Christine Solnon – p.6

Binary constraint satisfaction

☞ Using four parameters it is possible to make predictions about the difficulty of solving binary CSPs (Smith, 1993; Gent et al., 1996) ① the number of variables n ② the domain size m ③ the constraint density p1 ④ the average tightness of constraints p2

Jano van Hemert & Christine Solnon – p.6

A binary CSP: parameters

x1 x2 x3 x4

a b c a 1 b 1 1 c a b c a b 1 c 1 1 a b c a 1 b 1 c 1

Jano van Hemert & Christine Solnon – p.7

A binary CSP: parameters

x1 x2 x3 x4

a b c a 1 b 1 1 c a b c a b 1 c 1 1 a b c a 1 b 1 c 1

n, m, p1, p2 = 4, ?, ?, ?

Jano van Hemert & Christine Solnon – p.7

A binary CSP: parameters

x1 x2 x3 x4

a b c a 1 b 1 1 c a b c a b 1 c 1 1 a b c a 1 b 1 c 1

n, m, p1, p2 = 4, 3, ?, ?

Jano van Hemert & Christine Solnon – p.7

A binary CSP: parameters

x1 x2 x3 x4

a b c a 1 b 1 1 c a b c a b 1 c 1 1 a b c a 1 b 1 c 1

n, m, p1, p2 =

• 4, 3, 3

6, ?

• Jano van Hemert & Christine Solnon – p.7

A binary CSP: parameters

x1 x2 x3 x4

a b c a 1 b 1 1 c a b c a b 1 c 1 1 a b c a 1 b 1 c 1

n, m, p1, p2 =

• 4, 3, 1

2, 3 9

• Jano van Hemert & Christine Solnon – p.7

A binary CSP: parameters

x1 x2 x3 x4

a b c a 1 b 1 1 c a b c a b 1 c 1 1 a b c a 1 b 1 c 1

n, m, p1, p2 =

• 4, 3, 1

2, 1 3

• Jano van Hemert & Christine Solnon – p.7

Phase transition

✔ In general, a phase transition is defined as a change of a system ✔ In the context of solving NP-complete problems, it refers to a change in properties such as solvability and hardness of solving ✔ For BINCSP we shall use the connectivity as an order parameter ✔ Using the constrainedness we may predict the hardness of problem solving, this can be expressed as

κ = n − 1 2 p1logm( 1 1 − p2 )

Jano van Hemert & Christine Solnon – p.8

Three methodologies . . . . . . three algorithms

Jano van Hemert & Christine Solnon – p.9

Ant Colony Optimisation

✔ Ants create a path that corresponds to an assignment

• f the variables (Solnon, 2002)

✔ The order of assigning variables is set by the smallest domain heuristic ✔ Post-processing with a local search where at random variables are chosen that are involved in conflicts; their assignments are changed to minimise the number of conflicts (Minton et al., 1992) ✔ Pheromone trails are updated according to the best assignment found

Jano van Hemert & Christine Solnon – p.10

Evolutionary Computation

✔ Glass-Box (Marchiori, 1997), one of the three best algorithms from an extensive comparison study (Craenen et al., 2003) ✔ Employs a repair method that tries to change the variable assignments of each constraint in order to get it satisfied ✔ Uses a heuristic; constraints with the most conflicts are changed first

Jano van Hemert & Christine Solnon – p.11

Constraint Programming

✔ Forward checking with conflict-directed Backjumping (Prosser, 1993) ✔ Forward checking decreases the domains of unassigned variables and moves forward until a solution

• ccurs or until a domain becomes empty

(Haralick and Elliot, 1980) ✔ Conflict-directed backjumping improves the speed by making larger jumps during backtrack moves (Dechter, 1990) ✎ The method is complete, and will always provide a solution if it exists

Jano van Hemert & Christine Solnon – p.12

Experiments

Jano van Hemert & Christine Solnon – p.13

Measurement: success ratio

✔ Number of successful runs divided by total runs ✔ Every problem instance used in the experiments has at least one solution ✔ Note that FC-CBJ is always successful, i.e., the success ratio is one

Jano van Hemert & Christine Solnon – p.14

Measurement: search effort

✔ The effort to search for a solution is commonly measured in the number of conflict checks ✔ One conflict check occurs when an algorithm checks whether the assignment of values to two variables is forbidden or not ✔ This measurement is not influenced by hardware or software issues

Jano van Hemert & Christine Solnon – p.15

Measurement: resampling ratio

✔ Let S denote the set of points in the search space sampled by an algorithm ✔ Then, define resampling ratio as # evaluations − |S| # evaluations (van Hemert and Bäck, 2002) ✔ The resampling ratio provides a means of observing how efficient an algorithm is sampling the search space; a high value corresponds to a low efficiency

Jano van Hemert & Christine Solnon – p.16

Experiments

Two experiments: ① Phase transition: n = 20, m = 20, p1 = 1, and 10 settings

• f p2: 0.22, 0.23, . . . , 0.28

② Scale-up: n ∈ {15, 20, . . . , 60}, m = 20, p1 = 1, and p2 is set to ensure a constrainedness of κ = 0.92

Jano van Hemert & Christine Solnon – p.17

Experiments

Two experiments: ① Phase transition: n = 20, m = 20, p1 = 1, and 10 settings

• f p2: 0.22, 0.23, . . . , 0.28

② Scale-up: n ∈ {15, 20, . . . , 60}, m = 20, p1 = 1, and p2 is set to ensure a constrainedness of κ = 0.92 ✎ For each setting 25 unique problem instances are used, generated using Model E (Achlioptas et al., 2001) using RandomCsp (download available from my home page)

Jano van Hemert & Christine Solnon – p.17

Experiments

Two experiments: ① Phase transition: n = 20, m = 20, p1 = 1, and 10 settings

• f p2: 0.22, 0.23, . . . , 0.28

② Scale-up: n ∈ {15, 20, . . . , 60}, m = 20, p1 = 1, and p2 is set to ensure a constrainedness of κ = 0.92 ✎ For each setting 25 unique problem instances are used, generated using Model E (Achlioptas et al., 2001) using RandomCsp (download available from my home page) ✎ For ACO and EC we perform 10 independent runs on each instance; for CP we perform one

Jano van Hemert & Christine Solnon – p.17

Phase transition: success ratio

0.2 0.4 0.6 0.8 1 0.22 0.23 0.24 0.25 0.26 0.27 0.28 success rate Instance tightness p2 Glass-Box Ant-Solver

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.22 0.23 0.24 0.25 0.26 0.27 0.28 resampling ratio Instance tightness p2 Glass-Box Ant-Solver

Jano van Hemert & Christine Solnon – p.18

Phase transition: conflict checks

10000 100000 1e+06 1e+07 1e+08 1e+09 0.22 0.23 0.24 0.25 0.26 0.27 0.28 conflict checks Instance tightness p2 Glass-Box Ant-Solver FC-CBJ

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.22 0.23 0.24 0.25 0.26 0.27 0.28 resampling ratio Instance tightness p2 Glass-Box Ant-Solver

Jano van Hemert & Christine Solnon – p.19

Scale-up: success ratio

0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 40 45 50 55 60 success rate number of variables Glass-Box 0.1 0.2 0.3 0.4 0.5 0.6 0.7 5 10 15 20 25 30 35 40 45 50 55 60 resampling ratio number of variables Glass-Box Ant-Solver

Jano van Hemert & Christine Solnon – p.20

Scale-up: conflict checks

10 100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09 1e+10 1e+11 5 10 15 20 25 30 35 40 45 50 55 60 conflict checks number of variables Glass-Box Ant-Solver FC-CBJ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 5 10 15 20 25 30 35 40 45 50 55 60 resampling ratio number of variables Glass-Box Ant-Solver

Jano van Hemert & Christine Solnon – p.21

Conclusions

Jano van Hemert & Christine Solnon – p.22

Conclusions

✔ Evolutionary algorithms are not up to the job of solving hard binary CSPs

Jano van Hemert & Christine Solnon – p.23

Conclusions

✔ Evolutionary algorithms are not up to the job of solving hard binary CSPs ✔ Ant-solver is able to compete with FC-CBJ, and has a better scale-up property

Jano van Hemert & Christine Solnon – p.23

Conclusions

✔ Evolutionary algorithms are not up to the job of solving hard binary CSPs ✔ Ant-solver is able to compete with FC-CBJ, and has a better scale-up property ✔ The resampling ratio is a good indication to show the inefficient sampling when algorithms have a poor performance

Jano van Hemert & Christine Solnon – p.23

References

Achlioptas, D., Kirousis, L., Kranakis, E., Krizanc, D., Molloy, M., and Stamatiou, Y. (2001). Random constraint satisfac- tion: A more accurate picture. Constraints, 4(6):329–344. Craenen, B., Eiben, A., and van Hemert, J. (2003). Compar- ing evolutionary algorithms on binary constraint satisfac- tion problems. IEEE Transactions on Evolutionary Compu- tation, 7(5):424–444. Dechter, R. (1990). Enhancement schemes for constraint pro- cessing: Backjumping, learning, and cutset decomposi-

• tion. Journal of Artifcial Intelligence, 41:273–312.

Gent, I. P ., MacIntyre, E., Prosser, P ., and Walsh, T. (1996). The constrainedness of search. In Proceedings of the AAAI-96, pages 246–252. Haralick, R. and Elliot, G. (1980). Increasing tree search effi- ciency for constraint-satisfaction problems. Artificial Intelli- gence, 14(3rd):263–313. Marchiori, E. (1997). Combining constraint processing and ge- netic algorithms for constraint satisfaction problems. In B¨ ack, T., editor, Proceedings of the 7th International Con- ference on Genetic Algorithms, pages 330–337, San Fran- cisco, CA. Morgan Kaufmann Publishers, Inc. 23-1

Minton, S., Johnston, M., Philips, A., and Laird, P . (1992). Minimizing conflicts: a heuristic repair method for con- straint satistaction and scheduling problems. Artificial In- telligence, 58:161–205. Prosser, P . (1993). Hybrid algorithms for the constraint satisfac- tion problem. Computational Intelligence, 9(3):268–299. Smith, B. (1993). The phase transition in constraint satisfaction problems: A closer look at the mushy region. Technical Report RR 93.41, University of Leeds. Solnon, C. (2002). Ants can solve constraint satisfaction prob- lems. IEEE Transactions on Evolutionary Computation, 6(4):347–357. Tsang, E. (1993). Foundations of Constraint Satisfaction. Aca- demic Press. van Hemert, J. and Bck, T. (2002). Measuring the searched space to guide efficiency: The principle and evidence on constraint satisfaction. In Merelo, J., Panagiotis, A., Beyer, H.-G., Fern´ andez-Villaca˜ nas, J.-L., and Schwefel, H.-P ., editors, Parallel Problem Solving from Nature — PPSN VII, number 2439 in Springer Lecture Notes on Computer Sci- ence, pages 23–32. Springer-Verlag, Berlin. 23-1