Building “ Problem Solving Engines ” for Combinatorial Optimization Toshi Ibaraki Kwansei Gakuin...
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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
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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
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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
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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.
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Approach by Standard Problems
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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)
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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
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Standard problem:
Constraint satisfaction problem (CSP)
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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
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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
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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.
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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
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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.
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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}
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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
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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
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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