Building “ Problem Solving Engines ” for Combinatorial Optimization Toshi Ibaraki Kwansei Gakuin University (+ M. Yagiura, K. Nonobe and students, Kyoto University) Franco-Japanese Workshop on CP, Oct , 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 discrete optimization 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 n 2 or n 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 X i and their domains D i m constraints C l equalities, inequalities, nonequalities (all-different), linear and nonlinear formulae Hard and soft constraints; weights w l given to constraints C l Minimization of total penalt ｙ p(X) =Σ w l p l (X) p l (X): penalties given to violations of C l

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 D ata structures to skip X i and their values having apparently no improvement (i.e. partial propagation) Evaluation function for the search q(X) =Σ v l p l (X) ( possibly v l ≠ w l ) Automatic control of weights v l Frequent violation of C l  larger v l 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 , K. Nonobe and T. Ibaraki, An improved tabu search method for the weighted constraint satisfaction problem, INFOR, Vol. 39, No. 2, pp , M. Fukumori, Tabu search algorithm for the quadratic constraint satisfaction problem, Master thesis, Kyoto University, 2004.

CSP: Case study Nurse 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 X ij （ nurse i, j-th day ） Domain D ij ={ D, E, N, M, OFF }

Nurse scheduling problem Constraints 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 study Social 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 X tj （ 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