Building“Problem Solving Engines”

for Combinatorial Optimization

Toshi IbarakiKwansei Gakuin University(+ M. Yagiura, K. Nonobe and students, Kyoto University)

Franco-Japanese Workshop on CP, Oct. 25-27, 2004

Approaches to general solvers

Attempts from artificial intelligence GPS (general problem solver), resolution principle, ..., CP (constraint programming)

Attempts from mathematical programming Linear, nonlinear, integer programming, ...

Problem solving engines for discreteoptimization problems

Complexity Theory Class NP

Contains almost all problems solvable by enumeration

NP-hard (NP-complete) 　　 SAT (satisfiability), IP (integer program), . . .

Two implications

2. No algorithm can solve IP in polynomial time

1. All problems in NP can be reduced to IP

Approach by Approximate Solutions

Approximate solutions are sufficient in most applications. NP-hard problems can be approximately solved in

polynomial time.

But . . .

1. Problem sizes may explode during reduction processes.

e.g. the number of variables may become n2 or n3.2. The distance to optimality may not be preserved. Good approximate solutions to IP may not be good

solutions to the original problem. Only “natural” reductions are meaningful.

Approach by Standard Problems

List of Standard Problems

Integer programming (IP) Constraint satisfaction problem (CSP) Resource constrained project scheduling

problem (RCPSP) Vehicle routing problem (VRP) 2-dimensional packing problem (2PP) Generalized assignment problem (GAP) Set covering problem (SCP) Maximum satisfiability problem (MAXSAT)

Approximation Algorithms

Efficiency, generality, robustness, flexibility, . . .

Can such algorithms exist?

Local search (LS)

Genetic algorithm, simulated annealing, tabu search, iterated local search, GRASP, variable neighborhood search, . . .

Yes!

Metaheuristics

Standard problem:

Constraint satisfaction problem (CSP)

CSP: Definition n variables Xi and their domains Di

m constraints Cl 　 equalities, inequalities, nonequalities (all-different), linear and nonlinear formulae

Hard and soft constraints; weights

wl given to constraints Cl Minimization of total penalt ｙ 　　　 p(X) =Σ wl pl(X)

pl(X): penalties given to violations

of Cl

Comparison with IP

Flexible forms of constraints　 Compact formulations with small

numbers of variables and constraints

Soft constraints and objective functions via penalty functions

Algorithms by metaheuristics　 Robust performance even for problems

not suited for IP

CSP Algorithm

Algorithm framework: tabu search Local search using shift neighborhood

Checks all solutions obtainable by changing the value of one variable 　　　　　

Tabu list Prohibits changing those variables whose values were modified in recent t iterations, where t is tabu tenure.

Improvements Reduction of the neighborhood size

Data structures to skip Xi and their values having apparently no improvement (i.e. partial propagation)

Evaluation function for the search 　　　　 q(X) =Σ vl pl(X) (possibly vl≠wl)

Automatic control of weights vl 　　　　　　

Frequent violation of Cl larger vl 　　　　　

　　　　　 Similar to subgradient method for Lagrangean multipliers

References for details

K. Nonobe and T. Ibaraki, A tabu search approach to the constraint satisfaction problem as a general problem solver, European J. of OR, Vol. 106, pp. 599-623, 1998.

K. Nonobe and T. Ibaraki, An improved tabu search method for the weighted constraint satisfaction problem, INFOR, Vol. 39, No. 2, pp. 131-151, 2001.

M. Fukumori, Tabu search algorithm for the quadratic constraint satisfaction problem, Master thesis, Kyoto University, 2004.

CSP: Case studyNurse scheduling problem

25 nurses （ Team A:13, B:12 ） Experienced nurses and new nurses 3 shifts （ day, evening, night ） ,

meetings, days off Time span ： 30 days

Formulation to CSP ：Variables Xij （ nurse i, j-th day ）Domain Dij ={D, E, N, M, OFF}

Nurse scheduling problemConstraints

Required numbers of shifts D, E, N in each day Upper and lower bounds on the numbers of shifts

and OFF’s assigned to each nurse in a month Predetermined M’s and OFF’s At least one OFF and one D in 7 days Prohibited patterns: 3 consecutive N; 4

consecutive E; 5 consecutive D; D, E or M after N; D or M after E; OFF-work-OFF

N should be done in the form NN; at least 6 days before the next NN

Balance between teams A and B Many others

CSP: Case studySocial golfer problem n golfers play once a week, always in m

groups, each consisting of n/m players. No two golfers want to play together more

than once. Find a schedule with the largest number

of weeks.

Formulation to CSP ：Variables Xtj （ t-th week, group j ）Domain D = power set of {i=1, 2, …, n}Nonlinear constraints

Future Directions Further improvement of metaheuristic

algorithms Increasing the formulation power of

standard problems Other standard problems Aggregation of all algorithms into a

decision support system User interfaces. Supports to model

application problems