GENET and Tabu Search for Combinatorial Optimization Problems Dr. J. F. Boyce, C. H. D. Dimitropoulos, G. vom Scheidt, Prof. J. G. Taylor Wheatstone Laboratory, King’s College London, Strand, London WC2R 2LS, U.K.
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Abstract. Constraint satisfaction problems are combinatorial optimization problems which involve finding an assignment to a set of variables that is consistent with a set of constraints. In this paper the use of two local search techniques, namely GENET and Tabu Search for constraint satisfaction optimization problems and partial constraint satisfaction problems is investigated. These methods are compared by application to a difficult partial constraint satisfaction optimization problem, namely the Radio Links Frequency Assignment Problem (RLFAP).
1 Introduction A constraint satisfaction problem consists of finding an assignment to a set of variables that is consistent with a set of constraints. In this paper we assume that the number of variables is finite and the respective domains are also finite and discrete. Constraints restrict the configurations which may be assigned to subsets of the variables. There are many real-world problems that can be described within this formalism, for example scheduling, labeling in vision, bin-packing, and frequency assignment problems. Constraint Satisfaction Problems: A CSP is a finite set V of variables, a function D which maps every variable to a set of domains, and a set C of constraints. Constraint Satisfaction Optimization Problems: In many real-world instances, it is not sufficient to find just any assignment which satisfies all constraints. If there is a multitude of valid solutions to a problem, i.e more than one assignment which violates no constraints, additional requirements on the minimization process may be introduced. In some circumstances one solution is as good as any other, but in scheduling or frequency assignment problems, solutions will vary in resource usage of some form and hence an optimal solution should also minimize these secondary costs. Constraint satisfaction optimization problems may be modeled as a special case of PCSPs, as defined below. Partial Constraint Satisfaction Problems: In some instances of constraint satisfaction problems, the nature of the problem may only require a subset of the constraints to be met, or may be so over-constrained that only a subset of the constraints can be met, i.e no complete solution exists. In either case a partial solution will be accepted, but an objective-function or cost-function may be introduced to grade these solutions with respect to secondary objectives, such as resource usage, or importance of constraints. A PCSP is a finite set of variables, a set of finite domains for the variables, a finite set of constraints on an arbitrary subset of variables, and an objective-function which maps every assignment for the set of variables to a scalar. An optimal solution to a PCSP is an assignment which satisfies a subset of the constraints and minimizes the objective-function. The objective-function may contain weights for violating certain constraints and any number of additional terms dependent on the variable assignments.
2 GENET and Tabu Search Combinatorial optimization problems and especially partial CSPs may be hard to tackle using complete methods like branch-and-bound due to the vast parameter space to be examined. Heuristic methods that produce acceptable approximate results in a short time and improve on them if more time is available are therefore an important alternative approach. Many of them are based upon a local search technique which employs some form of hill-climbing to find local minima, combined with an escape technique to try to leave their basins of attraction.
Genet and Tabu Search both fit into this paradigm and for our experiments both use a very similar local search method, while employing different escape techniques. During minimization, at each iteration a local neighbourhood (the set of states that can be reached by adjusting one variable) is examined and the resulting 1-optimal move is made. Both Tabu Search and Genet deform the cost surface as the search proceeds. Tabu search imposes hard limits on the possible search itineraries using the taboo list, while Genet uses a scheme of dynamic local penalization to fill up local minima and discourage incompatible assignments. Both approaches will be explained in the next sections.
2.1 2.1.1
GENET Introduction
Genet is a connectionist approach to CSPs developed by Tsang and Wang [1]. The problem representation in Genet derives from its neural network interpretation. A recurren t network with weighted inhibitory connections encodes the variables and constraints characterizing the problem instance. The network alternatingly settles into stable states corresponding to minima of the cost-surface and modifies connection strengths during the "learning" phase of the algorithm to escape from local minima. 2.1.2
Problem Modeling
Variables are represented by clusters of nodes with binary-valued activations. Each node of a cluster corresponds to one value in its variable’s domain, and only one node is active at any time, indicating the current assignment. For instance if a variable had a domain consisting of the values (10,20,30,40,50) and was assigned to a value of 40, the corresponding nodes would have activations (0,0,0,1,0). Constraints are modeled as inhibitory connections linking incompatible nodes. All connections are initially weighted with ‘-1’. During "learning" this inhibition can be strengthened, as the algorithm tries to discourage certain assignments. The weights can be stored as a n m matrix for each constrained variable-pair, where n and m are the respective domain sizes. As a generalization it can be assumed that all nodes are linked to all other nodes, with connections corresponding to unconstrained assignments set to a weight of ‘0’. The network is then fully connected. 2.1.3
Network Dynamics
Initially all variables are assigned to random values in their domains (one node in each cluster is set to one, all others to zero) and all weights connecting incompatible nodes are initialized to ‘-1’. Then the activation of all nodes is computed by summing over the weighted activations of all incoming connections,
aik(n+1) =
XjV j XjD j wijkln ajln l
( )
( )
l=1 j =1
(1)
where aik and ajl are activations of nodes, wijkl are the weights between them and jV j and jDl j are the number of variables and their domain sizes. Then in each cluster the node with the highest (least negative) input is activated and all others are inactivated. The active node represents the new assignment. This process is repeated until the network reaches a fixed point and no further updates occur. If the network has reached a global minimum, the algorithm stops. Otherwise Genet’s "learning" rule is applied and the algorithm tries to escape from the local minimum by imposing additional penalties on connections between active nodes corresponding to violated constraints: (n) (n) (n) (0 ) (n+1) = wijkl + aik ajl wijkl wijkl
(2)
Note that the product on the right-hand side will be zero for connections between unconstrained nodes and ‘-1’ if a constraint is violated, i.e both nodes are active and incompatible. 2.1.4
Extension to PCSPs and CSOPs
Genet can be described as a hill-climbing technique choosing 1-optimal moves in a cost-surface deformed by penalizations. The original formulation of Genet does not solve CSOPs or PCSPs. In Genet all constraints are treated as equal
and no secondary objectives are taken into account. But by weighting constraints and adding layers to the cost-surface that correspond to secondary objectives like resource usage, the original Genet has been extended to handle CSOPs and PCSPs [2]. (0) The input to a node can be separated into two terms: a constant ‘compatibility’ term wijkl which is ‘-1’ if nodes i and j of variables k and l are incompatible and ‘0’ otherwise, and a monotonically decreasing ‘inhibition’ term which contains the penalties that a constraint accumulates during the search. The cost-function then becomes
c(n) = ? 12
X jXD j X XjD j aikn ajln wijkl ? X jXD j X XjD j aikn ajln hijkln k
l
k2V i=1 l2V j =1
( )
( )
(0)
1 2
k
l
k2V i=1 l2V j =1
( )
( )
( )
(n) hijkl ,
(3 )
which yields the number of violations plus their penalties. By augmenting this cost-function with additional terms corresponding to secondary objectives and using weights proportional to the importance of constraints, the search is directed towards minima of a cost-function with these additional terms:
c(!x ) = ?W ? H (n) ? A(!x )
(4)
n
The law (1) then becomes gradient descent on (3) in the activity aik . Here W includes the ‘compatibility’ terms, H (n) includes the ‘inhibition’ terms (the penalties) and A(!x ) any additional terms, such as the number of frequencies ! used in a RLFAP, evaluated at position x . ( )
2.2
Tabu Search
Tabu Search is a modern heuristic method which was introduced by Glover [3, 4] as an efficient way of finding high quality solutions to hard combinatorial optimization problems. The name “Tabu” derives from the implementation of search guidance by the imposition of restrictions, either by direct exclusion of some moves or by modified probabilities for possible moves. The rules which govern the restrictions codify the a priori knowledge of the problem class. They depend both on the current structure of the neighbourhood of the solution and its history. The rules themselves may vary dynamically during the course of a search both in response to recent search behavior and to the number of search moves since initiation. The rules are based on the interaction between restriction and aspiration, where the former seeks to limit the moves available at any search state whilst the latter permits the overriding of restrictions if the search appears to be trapped in a local minimum. 2.2.1
Implementation
The RLFAP implementation of Tabu Search outlined below 1 requires three parameters to run: the size of the tabu list (jT j), the patience parameter (p), and the ratio parameter (N ). 1. Begin with a random assignment of values to variables. 2. Consider all variables within
N % of the maximum number of violations,
(a) Check all values for each variable, (b) Check for TABU and ASPIRATION. 3. Select the best move for the iteration & go to (2). 4. Store the best move so far and update tabu list 2 and aspiration criteria.
3
5. Repeat from step (2), until a solution is found or until there is no improvement in the cost for more than 1 For
p iterations.
a detailed description of this implementation see [5]. new move is added to the recency based tabu list, which means that for the next T iterations this move will be tabu. Recency based tabu was all that was required to find feasible assignments in most cases. However, on some occasions the same link would be repeatedly selected for assignment on each iteration and tabu search would cycle with different frequency assignments for the same link. To prevent this cycling and to diversify the search a frequency based tabu is imposed on that link for a fixed number of iterations. 3 A simple type of aspiration criteria is employed: a candidate move is licit (non-tabu) if the assignment yields a solution with a cost better than the best found so far. Hence, the updated aspiration value is A := min(A; cost 1).
j j
2 The
?
3 The Radio Links Frequency Assignment Problem We now assess the performance of GENET and Tabu Search for the Radio Links Frequency Assignment Problem (RLFAP), by application to a set of realistic instances, the CELAR problems 4 . Part of the work presented in this section is a result of the Combinatorial Algorithms for Military Applications (CALMA) project, where funding for King’s College London was provided by the DRA in the EUCLID Framework [6]. The radio links frequency assignment problem occurs in many civil and military applications [7, 8]. The main objective is to assign radio frequencies to a number of transmitters, subject to a number of constraints, so that a minimum of interference occurs. However, it may be impossible to satisfy all constraints, in which case trying to minimize the number of violated constraints (i.e. the interference) is a more realistic goal. The problem is NP-complete, and is a variant of the general T ?graph coloring problem, as introduced by Hale [9].
3.1
The CELAR data set of problems
In the CELAR problems, a set of possible frequencies is given to operate the radio links. Each of the radio links has to be assigned to one of the frequencies, while satisfying a usually large number of constraints of the following type: for any two "neighbouring" links i and j , if fi (resp: fj ) is the frequency assigned to link i (resp: to link j ) then jfi ? fj j > cij for ‘Inequality’ constraints, and jfi ? fj j = cij for ‘Equality’ constraints. The concept of "neighbouring" links, here, not only depends on the fact that the links lie geographically close to each other, but also on various electromagnetic characteristics (such as propagation, transmission power, etc : : : ) which may result in electromagnetic incompatibility (at certain frequencies) between the links. In some of the eleven CELAR scenarios there are many solutions and hence all constraints have to be met while minimizing resource usage (scen01–scen05,scen11), and in others there are no solutions and certain constraints have higher priority than others combined with restrictions on assignment mobility (scen06–scen10). So the CELAR data contains examples for CSPs, PCSPs, and CSOPs of varying size and difficulty. Each variable(link) can be assigned to a frequency from its respective domain of frequencies. The optimization criteria are:
scen01–03: multiple solutions, minimize the number of distinct frequencies used. scen04: multiple solutions, minimize the frequency span. scen05: multiple solutions, minimize the highest frequency used. scen06–08: no solutions satisfying all constraints, but a subset of hard constraints has to be satisfied and soft constraints have varying penalties for violation. scen09,10: as above, but variables also have varying mobility and reassignment of variables incurs varying penalties. scen11: multiple solutions, no secondary objectives.
3.2
Results
The following results for the CELAR data set were obtained using a C-implementation of Tabu Search and a C++implementation of the extension of Genet, both running on a 130 MHz DEC Alpha. Genet was optimized to use a minimal-neighbourhood change propagation scheme to accelerate the local search. Both Tabu Search and Genet are able to find good solutions to problems scen01–03. Note that the Genet results given for these problems are sample values, since they are subject to a time/quality tradeoff. Problems scen04 and scen05 are more tightly constrained and consequently much harder to solve. Here both methods benefit significantly from a pre-processing stage which exploits arc-consistency. Across all instances to which they were applied, both methods also perform well in comparison with other search techniques investigated in the Calma project. 4 The
data was provided by the French "Centre d’Electronique de l’Armement"(CELAR).
CELAR scen01 scen02 scen03 scen04 scen04arc scen05 scen05arc scen06 scen07 scen08 scen09 scen10 scen11
variables/ constraints 916/5548 200/1235 400/2760 680/3968 680/3968 400/2598 400/2598 200/1322 400/2865 916/2744 680/4103 680/4103 680/4103
best known cost 16 14 14 46 46 792 792 3623 374705 293 15716 31517 0
best found TABU GENET 18 16 14 14 14 14 n.a. 46 46 46 n.a. 792 792 792 9180 3852 6541695 435132 1745 366 16873 n.a. 31943 n.a. 0 0
found optimum TABU GENET n.a. 20% 70% 100% 20% 10% n.a. 100% 100% 100% n.a. 30% 40% 100% — — — — — — — — — — 60% 60%
average time TABU GENET 3hrs 75s 4min 9s 34min 32s n.a. 12s 10s 0.24s n.a. 8min 7min 2s 14min 10min 46min 18min 6hrs 32min 18min n.a. 2hrs n.a. 54min 25s
4 Conclusions Both methods are effective and flexible ways for solving partial constraint satisfaction and combinatorial optimization problems as present in the Celar set of Radio Link Frequency Assignment Problems. Genet appears to be much faster than Tabu Search. However, some of this may be due to its extensive optimization and hence it is difficult to directly compare the processing times. Experiments on small but tightly constrained frequency assignment problems for which Tabu Search almost always finds the optimal solution while Genet consistently fails, indicate that there may be a class of hard problems to which Genet is not applicable in the current implementation. Both methods were also tested for their behavior under perturbations of the problems, which occur if additional constraints or variables are introduced or existing ones removed. Since they employ local search and use a dynamic escape technique, they perform well under perturbations and can benefit from any previously found solutions. For their financial support, CHD would like to thank the DRA, and GvS would like to thank the German National Scholarship Foundation.
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