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Robust taboo search for the quadratic assignment problem

This paper proposes a robust form of Taboo Search for the Quadratic Assignment Problem (QAP) that can obtain solutions comparable to the best previously found without requiring multiple passes and altered parameter values. The method is efficient and succeeds in obtaining sub-optimal solutions for problems up to size 64.

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Robust taboo search for the quadratic assignment problem

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  1. Robust taboo search for the quadratic assignment problem Taillard, E., "Robust taboo search for the QAP," Parallel Computing, Vol. 17, pp. 443-455, 1991.

  2. 1. Introduction • For more than 30 years researches have been proposing exact and heuristic methods for the quadratic assignment problem (QAP). • This problem is so hard that an instance of size 36 proposed in 1961 by Steinberg has not been solved yet.

  3. For bigger sizes, heuristic methods must be used. • Although obtaining the best results reported in the literature, TS has been applied in this setting by performing multiple solution passes for each problem, starting from different initial solutions and modifying parameters between the solutions.

  4. The goal of this paper is to propose a more robust form of TS for QAP that has a minimum number of parameters and is very easy to implement, capable of obtaining solutions comparable to the best previously found without requiring multiple passes and altered parameter values. • Our method is efficient and succeeds in obtaining sub-optimal solutions (conjectured but not proved to be optimal) for problems up to size 64.

  5. 2. The quadratic assignment problem

  6. Examples of problems that may naturally be expressed in terms of QAP includes: • The placement of logical modules in a chip such that the total length of the connections is minimized. • The distribution of medical services in a large hospital centre. • …

  7. The QAP contains the traveling salesman problem as a special case and hence is NP-hard. • However, the asymptotic behavior of randomly generated problem, under some assumptions, shows that the relative difference between the best and the worst solutions tends to zero if the problem size tends to infinity.

  8. 3. Tabu Search • The first ingredient is to define a neighborhood, or a set of moves that may be applied to a given solution to produce a new one. • Among all neighboring solutions, TS seeks one with a best heuristic evaluation. • In simplest case such an evaluation dictates the choice of a move that improves most the objective function.

  9. If there are no improving moves, TS chooses one that least degrades the objective function. • In order to avoid returning to the local optimum just visited, the reverse move now must be forbidden. This is done by tabu list. • This list contains a number s of elements defining forbidden (taboo) moves; the parameter s is called the taboo list size. • An aspiration criterion is introduced to allow taboo moves to be chosen if they are judged to be interesting.

  10. 3.1 Moves • A natural type of move that is very well suited for this problem consists of exchanging two units so that each occupies the location formerly by the other. • Starting from a placement, a neighboring placement πis obtained by permuting units r and s:

  11. Value of Move

  12. 3.2 Taboo List • Skorin-Kapov has proposed to forbid the exchange of two units that were exchanged during the preceding s iterations. • The taboo list is constituted of pairs (i, j) of units that cannot be exchanged. • By this means, testing whether a move is taboo or not requires only one comparison.

  13. 3.3 Aspiration function

  14. 3.4 Taboo list size • The choice of taboo list size is critical; if its value is too small, cycling may occur in the search process while if its too large, appealing move may be forbidden and leading to the exploration of lower quality solutions, producing a larger number of iterations to find the solution desired. • Rather than fixing the size s at a constant value, we propose to change this value randomly during the search.

  15. Practically, S will be chosen between Smin and Smax=Smin + Δ and frequently changed, for example every 2×Smax iterations. • In fact, for most problems we found the best solutions known up to now using Smin =[0.9 N] and Smax=[1.1 N].

  16. 4. Random problems

  17. 5. Parallel taboo search • Without changing the sequential algorithm significantly, the most efficient way of executing a TS concurrently is to distribute on many processors the component that requires the most CPU time.

  18. The number of iterations will effect the solutions. • In Figure 4, every problem is run 300 times to obtain the best known solution.

  19. 6.Numerical results

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