A heuristic for solving the frequency assignment problem

Transcription

1 A heuristic for solving the frequency assignment problem Mireille Palpant, Christian Artigues and Philippe Michelon Laboratoire d Informatique, Université d Avignon, France. mireille.palpant@lia.univ-avignon.fr, christian.artigues@lia.univ-avignon.fr, philippe.michelon@lia.univ-avignon.fr Abstract We present a large neighbourhood search method for solving the frequency assignment problem with multiple source interferences. Starting from an initial solution, subproblems of auto-adjustable size are iteratively generated and solved with either a greedy algorithm or an exact method. The approach has been successfully tested on realistic instances issued from military applications. Keywords: frequency assignment, heuristics, large neighbourhood search.

2 1. Introduction In combinatorial optimisation, some local search methods aim at exploring large neighbourhoods by implicit enumeration methods. For the quadratic assignment problem, the Mimausa method designed by Mautor and Michelon [3], build at each iteration a reduced problem and solves it by branch and bound. For the vehicle routing problem, Shaw [5] considers the removal of several customer visits and re-insert them by using Limited Discrepancy Search (LDS). Caseau et al [2] have also successfully applied the so-called Forget and Extend heuristic on the job-shop problem, freezing a set of tasks at each iteration and solving the remaining sub-problem by LDS associated with constraint propagation techniques. In the same spirit, Palpant et al [4] have proposed in a previous work an efficient large neighbourhood search technique for the resource-constrained project scheduling problem. In this paper, we propose a local search method based on [4] to solve the minimum interference frequency assignment problem (MI-FAP). We first check whether the problem is feasible, then try to minimize the span (i.e. the difference between the minimum and maximum used frequency). At each iteration a subproblem is derived from the current solution and solved with the help of constraint propagation techniques. The resolution can be performed by different methods depending on the resolution time and solution quality desired. Thus the use of a greedy algorithm provides medium quality solutions in a very short time while a truncated exact method obtains better solutions spending more time in the search. In the approach developed we have focused on the combination of these two methods in a local search scheme. We have also emphasized the sub-problem generation, leaving its resolution to a commercial constraint programming solver. 2. Problem description The considered approach proposes to solve the frequency assignment problem in hertzian networks. Such networks consist of a collection of sites where antennas related to emitters and receivers are located. A connection establishes a link between two distinct geographic sites and can be composed by several transmission channels. Each channel i = 1,, n possesses emission and reception characteristics given by a system σ(i). It must be assigned to a pair (f i, w i ), where f i represents a frequency of its domain D ϕ(i) and w i the wave polarisation. The present work only considers the case of a unique polarisation possibility. Consequently the assignment problem amounts to find a n-tuple F = {f 1,, f n } which satisfies (partially) the following constraints: - Simple Equality constraints: f i = f j, (i,j) SE; - Simple Inequality constraints : f i f j, (i,j) SI; - Duplex Equality constraints: f i f j = ε ij, (i,j) DE; - Duplex Inequality constraints f i f j ε ij, (i,j) DI; - Co-site constraint: f i f j δ ij, (i,j) C; - Global constraints: j i λ ij T σ(i) σ(j) fi f j Λ i, i G. The tabulated non-negative functions T σ(i) σ(j), defined for all possible pairs of systems, are calculated from selectivity and normalized spectre curves and decrease as the gap between frequencies i and j increases.

3 Simple and duplex constraints SE, SI, DE and DI are grouped together under the term of imperative equality and inequality constraints IE and II and can be all formulated as duplex constraints with a possibly null ε ij. They have strictly to be satisfied. Duplex constraints impose or forbid a frequency gap on a pair of channels belonging to the same connection. Co-site and global constraints C and G belong to electro-magnetic constraints. They are used to model interferences between channels. To avoid any interference between a perturbed and a disruptive channel, whose respective receiver and emitter are located on the same site, co-site constraints C impose a minimal gap on the two assigned frequencies. In the same way, global constraints G enable to take into account the case of several disruptive channels associated with emitters not collocated with the perturbed channel receiver. In classical approaches the disruptive right is evenly distributed among all disruptive channels and modelled by co-site type binary constraints. To provide a model closer to practical situations, we replace here each set of binary constraints by a single global constraint. This new global approach is also less restrictive. This may hopefully lead to some improvement in the solution quality for both interference and span criterions. The objective of the problem is double: on the one hand it consists in minimizing the level of interference given by the weighted sum α V + β W of respectively co-site and global violated constraints until a feasible solution is found; on the other hand, it tends to minimize the span if the previous objective has been reached. Being strongly NP-hard, the FAP has been widely studied over the last years. We refer to [6] for a recent survey on this problem. 3. Overview of the method 3.1. The main algorithm The main algorithm, which is directly adapted from the method presented in [4], runs as described below. The procedure stops when the total execution time reaches the requested execution time. The first phase (interference minimization) runs until the number of violated constraints becomes null. 1. pre-processing: 1.1. reduction of channels domains due to imperative equality constraints : for each IE such that f i f j = ε ij for each value v of D ϕ(i) (respectively D ϕ(j) ) if v ε ij and v + ε ij D ϕ(j) (respectively D ϕ(i) ) then remove v from D ϕ(i) (respectively D ϕ(j) ) 1.2. problem decomposition in k connex components: CC 1,, CC k of size n 1,, n k, respectively 2. generation of an initial solution S 0 : for i = 1 to k 2.1. apply the greedy_init heuristic to CC i

4 3. interference minimisation: 3.1. greedy phase repeat select the next connex component CC i such that at least one constraint is violated select p i n i channels in A s and generate the associated subproblem SPI s solve SPI s with the greedy_iter heuristic eventually adjust the subproblem size p i until tot_time = 3/4*req_time or nb_violated = exact phase repeat select the next connex component CC i such that at least one constraint is violated select p i n i channels in A s and generate the associated subproblem SPI s solve SPI s with the exact method in maximum exact_time time until tot_time = req_time or nb_violated = 0 4. span minimisation: repeat 4.1. select the next connex component CC i 4.2. select p i n i channels in A s and generate the associated subproblem SPS s 4.3. solve SPS s with the exact method in maximum exact_time time until tot_time = req_time In the next sections, we describe how each of the main steps of the algorithm is performed The greedy_init heuristic For the generation of an initial solution, we use a simple greedy method. At each iteration, greedy_init selects the most constrained channel of smallest domain and assigns it to the lowest frequency possible that minimizes the weighted sum of electromagnetic violated constraints (C+G). We perform two executions of this algorithm with a different definition of the channels constraint degree each time. During the first execution, the degree of a channel is given exactly by its number of occurrences in any constraint. During the second one, an occurrence in a global constraint of a channel as a disrupter increases the degree only by Selection of the subproblem Let p i denote the current number of channels to be included in the subproblem of connex component CC i at the current iteration s. We initialise p i to min(15, n i 1) where n i is the total number of channels in CC i. We compute the set A s as follows: a start channel x is selected randomly among the channels that belong to at least one violated constraint in the current solution. Then any channel i linked with x by any constraint is included in A s, in the order given by a randomly mixed index list L. We

5 repeat the process recursively from the next selected channel until p i channels are included in A s. The interest of the index list lies in bringing some diversification to the selection process. Indeed, starting from a given channel, it enables to obtain different subproblem selections. The example below shows an execution of the selection process for the connex component CC 0. At each iteration we compute a sub-graph G x which associates an edge (i, x) with a possible selection of the channel i from the channel x. In the adopted selection method, an edge (i, x) belongs to G x if the two channels i and j are linked together by any duplex, co-site or global constraint. n 0 = 18, p 0 = 15 L = {10, 5, 4, 8, 15, 11, 2, 0, 12, 3, 14, 7, 6, 16, 17, 1, 13, 9} x A s = {2, 10, 8, 11, 0, 3, 14, 16, 17} L = {10, 5, 4, 8, 15, 11, 2, 0, 12, 3, 14, 7, 6, 16, 17, 1, 13, 9} x A s = {2, 10, 8, 11, 0, 3, 14, 16, 17}

6 L = {10, 5, 4, 8, 15, 11, 2, 0, 12, 3, 14, 7, 6, 16, 17, 1, 13, 9} x A s = {2, 10, 8, 11, 0, 3, 14, 16, 17, 5, 4, 15, 12, 7, 6} We have also considered other selection methods whose efficiency has been tested. The table below displays the average results obtained over thirty instances (that we will present in more details in section 4) in term of solution quality (both average objective function value and span) and time requirement by the following methods : - total random selection (S rand ); - selection of channels linked by imperative (or co-site or global) constraints only (S ic_only or S cc_only or S gc_only ); - retained selection (S dc_cc_gc ). Table 1. Comparison of different selection methods S rand S ic_only S cc_only S gc_only S ic_cc_gc Objective Span CPU This table clearly shows the domination of S ic_only and S ic_cc_gc selections in term of objective function value over the other methods while all methods are quite comparable if we refer to the span criterion. If we detail the obtained results we can remark the good behaviour of S rand on small and easy instances on the contrary of S cc_only and especially S gc_only which give very bad quality solutions on almost all instances. Finally we have chosen the S ic_cc_gc method which is less efficient than S ic_only because of its inferior time requirement Generation of the subproblem Depending on the objective, we have two kinds of subproblems to tackle: SPI for interference minimization, SPS for span minimization. At the current iteration s SPI s or SPS s are obtained by constraining any channel i not belonging to A s to be assigned to its current value f i s : f i = f i s, i A s. Then, each channel i A s is associated with a reduced domain induced by imperative and co-site constraints that have to be satisfied computed as follows:

7 - redd i = D ϕ(i) : - {f k, f k f j s ε ij, f k f j s + ε ij, f i f j = ε ij, j A s } - { f j s ε ij, f j s + ε ij, f i f j = ε ij, j A s } - {f k, f k > f j s δ ij, f k < f j s + δ ij, f i f j δ ij, j A s } (SPS s only) The constraints of the global problem that belong also to the subproblem are then generated: - f i f j = ε ij, i, j A s (IE) - f i f j ε ij, i, j A s (II) - f i f j δ ij, i, j A s (type 1 C) - f i f j s + δ ij OR f i f j s δ ij, i A s, j A s (type 2 C: SPI s only) - j i, j A s λij T σ(i) σ(j) fi f j Λ i k i, k A s λik T σ(i) σ(k) fi f k s, i A s (type 1 G: first member possibly null) - j i, j A s λij T σ(i) σ(j) fi s f j Λ i k i, k A s λik T σ(i) σ(k) fi s f k s, i A s (type 2 G: first member imperatively not null) Finally the objective is defined in order to improve the current solution: - α V + β W < current_objective_value or - max(f) min(f) < current_span 3.5. Resolution of the subproblem For solving the subproblem, we use either a simple greedy method or a truncated exact method, both associated with constraint propagation techniques. The choice of the resolution method depends on the kind of subproblem to treat and the strategies considered. The principle of the greedy_iter algorithm is to select iteratively the channel of smallest domain and to assign it to the lowest frequency that minimizes the weighted sum of electro-magnetic violated constraints. This algorithm obviously has no guarantee to find a neighbour solution in which case the current solution remains unchanged. However it enables to quickly improve an average quality solution. Hence we use it first during the interference minimization phase to obtain a "sufficiently good" solution. The tree search exact method is also used to find the neighbour solution. The branching scheme consists in choosing the channel of smallest domain. The search stops as soon as a better solution is found (first-fit) or when the exact_time time limit is reached. This method is used to intensify the search procedure during the interference minimization phase by visiting several neighbour solutions. It is also used to solve SPS subproblems. In both methods every time a channel is assigned to a frequency, default propagation techniques (arc consistency AC5) are employed to reduce the domain of remaining channels.

8 3.6. Building the next current solution If a solution to the subproblem has been found, the solution S s+1 is derived from S s by setting all assignment of channels in A s to the value found for the subproblem, and by keeping the values in S s for the other channels. Note that by construction S s+1 is feasible Adjustment of the subproblem size In the greedy phase, if the best solution has not been improved for a given number max_no_imp of iterations, the subproblem size p i is increased in one if p i < n i 1 or set to its initial value otherwise. We have set empirically max_no_imp = Computational results The proposed method has been assessed on the FAPPG realistic instances provided by the CELAR in the context of military applications. We have coded the heuristics in C++ using ILOG tools (Concert and Solver) and the tests have been realized on a PC with 2 processors, 1.3 GHz and 1 GB RAM. We have performed the experiments with exact_time = 30s. The parameter req_time has been set to one hour. The results are displayed on table 2. For each instance, the number of channels, the span, the number of co-site (CC) and global (GC) violated constraints and the mean CPU time for obtaining the best solution are given. Table 2. Computational results on the CELAR instances Instance #Channels #Violated CC #Violated GC Span CPU (s) FAPPG /16 0/ FAPPG /25 0/ FAPPG /108 0/ FAPPG /94 0/ FAPPG /95 0/ FAPPG /284 0/ FAPPG /285 3/ FAPPG /1063 0/ FAPPG /2585 0/ FAPPG /608 0/ FAPPG /592 0/ FAPPG /3251 0/ FAPPG /1125 9/ FAPPG /486 0/ FAPPG / / FAPPG /136 30/ FAPPG /108 28/ FAPPG /130 0/ FAPPG / / FAPPG / / FAPPG /3259 0/ FAPPG /1898 0/

9 FAPPG /63 0/ FAPPG /102 0/ FAPPG /248 2/ FAPPG /316 0/ FAPPG /342 2/ FAPPG /111 0/ FAPPG /1444 0/ FAPPG /598 16/ The results appear to be good as well in term of interference minimization as span minimization: we have found nineteen instances feasible in a very short time (no more than five minutes) and for FAPPG1 and FAPPG23 we have obtained the optimal span. Similarly, on more difficult problems the number of violated constraints is relatively low, sometimes even optimal like for FAPPG25 and FAPPG27. The good behaviour of the method is particularly clear on the very big sized FAPPG15 for which we found eighteen violated constraints in about fifty minutes. However, in spite of these encouraging results, the method shows poor efficiency on FAPPG19. This could be explained by the evident difficulty of the problem (highest "number of constraints / n" ratio) or a bad choice of objective function weights. We have also tested our program on some FAPPI instances in which each global constraint of corresponding FAPPG instance is replaced by a set of co-site constraints, which is more constraining. Our goal is to weigh up the impact of global constraints on the quality of solutions. The results obtained are reported in the table below. Table 3. Computational results on the FAPPI CELAR instances Instance #Channels #Violated CC Span CPU (s) FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI / FAPPI /

10 We can clearly note that the number of feasible instances decreases (9 versus 12 for the 20 corresponding FAPPG problems). Furthermore when an instance is found feasible the span is always bigger than the one of the corresponding FAPPG problem. However, in that case, we can remark that the computational time is lower than on FAPPG, which underlines the difficulty to take global constraints into account. Nevertheless the results obtained on the two different types of problems tend to prove the validity of the new approach. 5. Concluding remarks and further work We have proposed a heuristic for solving a new approach of the frequency assignment problem inspired by realistic features of military applications. We have emphasized the usefulness of global constraints which enable to increase the solution quality for both interference and span criterions. The method is competitive in term of time requirement and obtains good results on a set of thirty instances. It would be interesting to know how much the CPU times and the quality could be improved by refining the subproblem resolution methods with more powerful and dedicated constraint propagation techniques, implementation of LDS, etc. Moreover other way of generating subproblems, particularly in the span minimization context, could also be promising. References [1] Adams, J., Balas, E., and Zawack, D. (1988). The Shifting bottleneck procedure for job shop scheduling, Management Science, 34, [2] Caseau, Y., Laburthe, F., Le Pape, C., and Rottembourg, B. (2001). Combining local and global search in a constraint programming environment, Knowledge Engineering Review, 16, [3] Mautor, T. and Michelon, P. (1997). Mimausa: a new hybrid method combining exact solution and local search. Second International Conference on Meta- Heuristics, MIC 2, Sophia Antipolis, France. [4] Palpant, M., Artigues, C. and Michelon, P. (2001). Solving the resourceconstrained scheduling problem with large neighbourhood search, LIA technical report, 255. [5] Shaw, P. (1998). Using Constraint Programming and Local Search Methods to Solve Vehicle Routing Problems, In proceedings Principles and Practice of Constraint Programming, CP-98, Pisa, Italy, Lecture Notes in Computer Science, 1520, [6] Aardal, K. I., Van Hoesel, S. P. M., Koster, A. M. C. A., Mannino, C., Sassano, A. (2001). Models and solution techniques for frequency assignment problems, ZIB- Report, 01-40