Dameng Deng | Jun Ma and Hao Shen
A Simple and Efficient Tabu Search Heuristics for Kirkman Schoolgirl Problem
TUCS Technical Report No 704, June 2005
A Simple and Efficient Tabu Search Heuristics for Kirkman Schoolgirl Problem Dameng Deng Turku University, Department of Mathematics and Turku Center for Computer Science 20014 Turku, Finland
[email protected]
Jun Ma and Hao Shen
Shanghai Jiao Tong University, Department of Mathematics 200030,Shanghai, PRC {mj904, haoshen}@sjtu.edu.cn
TUCS Technical Report No 704, June 2005
Abstract The Kirkman Schoolgirl Problem is the most famous historical instance of the Social Golfer Problem, which has been studied extensively by the constraint community in recent years because of its high symmetry property. In this paper, we present a simple Tabu search approach for the Kirkman Schoolgirl Problem. The approach is based on a formulation of the problem as a Constraint Satisfaction Problem(CSP). The algorithm is very efficient and can compute a solution for the problem in less than one second. Moreover, it can be used to handle other instances of Social Golfer Problem.
Keywords: Social Golfer Problem, Kirkman Schoolgirl Problem,Constraint Satisfaction Problem, Tabu search algorithm
TUCS Laboratory Discrete Mathematics for Information Technology Laboratory
1
Introduction
Highly symmetric problem are always challenging for Constraint Programming. As a benchmark example, the Social Golfer Problem has been studied extensively by the constraint community in recent years. The research has been focused on modelling the problem efficiently as a constraint satisfaction problem, and removing symmetries among solutions to prune as much as possible the subsequent search tree. In a recent paper[1], the authors applied some sophisticated symmetry breaking techniques called SBDD+ to search Kirkman’s Schoolgirl Problem. The techniques they used were efficient and as a result, they found all the seven unique solutions in a few seconds. But their techniques have some disadvantages: it is not simple and needs further refinement in order to be applied for other large instances of social golfer problem. In this paper, we are also interested in Kirkman Schoolgirl problem. We present a simple local search approach based on tabu search which can be used to find a solution for the problem in less than one second. The remaining of the paper is organized as follows: we first define the Social Golfer Problem and its special instance, Kirkman Schoolgirl Problem. After modelling the problem as a constraint satisfaction problem, we present the Tabu search algorithm and our computational results. We mention the disadvantage of our method and give some concluding remarks in the end.
2 Problem Description and CSP Formulation The following is the description of the Social Golfer problem: • There are |P | = g × s golf players, where P = {p1 , p2 , · · · , pgs } is the set of all golf players. • Every week, all the players are divided into g groups, with s players in each group. Players within the same group play each other. • Any two players play at most once in the same group. Now the question is: how many weeks can this be arranged? Example 2.1 The following is a 6 weeks arrangement of 24 players into 6 groups, with 4 players in one group.For simplicity, we use {0, 1, · · · , 23} to denote player set. week1 : {14, 1, 10, 9}, {22, 4, 2, 5}, {6, 19, 8, 15}, {0, 20, 21, 23}, {12, 7, 17, 11}, {3, 18, 16, 13} week2 : {17, 1, 8, 5}, {0, 10, 15, 13}, {12, 2, 21, 14}, {4, 3, 7, 9}, {18, 6, 11, 23}, {19, 16, 20, 22} week3 : {17, 13, 20, 9}, {4, 16, 10, 6}, {22, 23, 14, 8}, {11, 3, 21, 1}, {12, 15, 5, 18}, {2, 19, 7, 0} week4 : {1, 4, 20, 15}, {8, 13, 21, 7}, {11, 19, 5, 10}, {6, 12, 22, 3}, {16, 2, 23, 9}, {18, 0, 14, 17} 1
week5 : {8, 0, 12, 16}, {3, 17, 10, 23}, {5, 7, 14, 20}, {1, 6, 2, 13}, {15, 22, 9, 11}, {4, 19, 18, 21} week6 : {9, 6, 0, 5}, {13, 4, 11, 14}, {18, 10, 22, 7}, {23, 1, 19, 12}, {17, 16, 15, 21}, {2, 20, 8, 3} Even though the description of the social golfer problem appears fairly easy, computational approaches have great difficulties solving even small instances in a reasonable amount of time. Note that the social golfer problem contains a remarkable number of symmetries. It is easy to see that players can be placed at any position within a group, groups can be rearranged within their week, and the weeks can be ordered arbitrarily. Furthermore, the players names can be permuted in any desired way. Thus the symmetry detecting and removing plays a key role during the search process in constraint programming. If there exists a w weeks arrangement, then we say it is a w − g − s solution. It is easy . The most famous historical instance of the social golfer problem to see that w ≤ gs−1 s−1 is the Kirkman Schoolgirl Problem(KSP), in which g = 5,s = 3 and there exists a 7 weeks arrangement. It was solved by Kirkman more than 150 years ago! But it is still an interesting testing problem for different models to find a solution in reasonable amount of computational time. To model the problem, we first introduce the Constraint Satisfaction Problem(CSP). A CSP [7] is defined by a triplet (X, D, C) with: • A finite set X of n variables: X = {X1 , · · · , Xn }; • A set D of associated domains: D = {D1 , · · · , Dn }. Each domain Di specifies the finite set of possible values of the variable Xi ; • A finite set C of p constraints: C = {C1 , · · · , Cp }. Each constraint is defined for a set of variables and specifies which combinations of values are compatible for these values. Given such a triplet, the problem is to generate a complete assignment of the values to the variables, which satisfies all the constraints: such an assignment is said to be consistent. Since the set of all assignment is defined by the Cartesian product D1 × · · · × Dn of the domains, solving a CSP means to determine a particular assignment among a potential huge number of possible assignment. The CSP model is powerful and general enough to model some well-known problems such as satisfiability, graph coloring as well as many practical applications. Now we form the Social Golfer Problem into a CSP. In the context of constraint programming, different models for Golfer Player Problem have been proposed[9]. We adopt the integer set model as described in [1]. In this model, the variables are the groups themselves and constraints are expressed as operations on sets. The variables Pi,j , with i the index of weeks and j the index of groups, are subsets of the set of Golfer players P . Each of them contains exactly s elements. All the groups of the same week are disjoint and every pair of groups from different weeks share at most 2
one element. All these properties can be expressed as the following constraints: Pi,j ⊂ {P1 , · · · , Pn }, |Pi,j | = s,
1≤i≤w
(1)
1 ≤ i ≤ w,
1≤j
