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The Gestalt heuristic: emerging abstraction to improve combinatorial search

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Abstract

Nowadays, many engineering applications require the minimization of a cost function such as decreasing the delivery time or the used space, reducing the development effort, and so on. Not surprisingly, research in optimization is one of the most active fields of computer science. Metaheuristics are part of the state-of-the-art techniques for combinatorial optimization. But their success comes at the price of considerable efforts in design and development time. Can we go further and automate their preparation? Especially when time is limited, dedicated techniques are unknown or the tackled problem is not well understood? The Gestalt heuristic, a search based on meta-modeling, answers those questions. Our approach, inspired by Gestalt psychology, considers the problem representation as a key factor of the success of the metaheuristic search process. Thanks to the emergence of such representation abstraction, the metaheuristic is being assisted by constraining the search. This abstraction is mainly based on the aggregation of the representation variables. The metaheuristic operators then work with these new aggregates. By learning, the Gestalt heuristic continuously searches for the right level of abstraction. It turns out to be an engineering mechanism very much related with the intrinsic emergence concept. First, the paper introduces the approach in the practical context of combinatorial optimization. It describes one possible implementation with evolutionary algorithms. Then, several experimental studies and results are presented and discussed in order to test the suggested Gestalt heuristic implementation and its main characteristics. Finally, the heuristic is more conceptually discussed in the context of emergence.

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Notes

  1. An approximate algorithm runs in provable polynomial time in the length and outputs a suboptimal solution.

  2. A heuristic usually, but not always, gives a good solution in a reasonable amount of time.

  3. A metaheuristic is a generic heuristic method combining black-blox procedures, usually heuristics themselves.

  4. A step walk being one stage of application of the operators of the given metaheuristics.

  5. Simple genetic algorithm is a genetic algorithm (GA) that evolves non-overlapping populations of binary encoded individuals. The operators are the one point crossover, the flip mutation and a fitness proportionate selection (Mitchell 1996; Goldberg 1989).

  6. The lens \(\mathcal{G}\) is responsible for mapping the original search space S(π) to \(S_{\Upgamma}\) with a one-to-one correspondence.

  7. Or the fitness value, in EA terminology.

  8. In EA, it corresponds to loci.

  9. http://www.research.att.com/~dsj/chtsp/download.html.

  10. http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/.

  11. See Appendix A for considerations about the gestalt description of a permutation.

  12. This racing algorithm tests different configurations, i.e. parameter sets, over different instances. For each instance, the given metaheuristic is run for each configuration. From the obtained results, the configurations are ranked with a Friedman test. Considering a given confidence level, the Friedman test gives the significantly bad configurations (i.e. with a corresponding p-value). These significantly worse configurations are deleted from the configurations pool and not considered in the next steps of the racing algorithm. An R package of Race is available on http://cran.r-project.org/src/contrib/Descriptions/race.html

  13. See Sect. 5.1.5.

  14. See Sect. 5.1.6.

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  1. IRIDIA, CoDE, Univeristé Libre de Bruxelles, Brussels, Belgium

    Christophe Philemotte & Hugues Bersini

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  1. Christophe Philemotte
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Correspondence to Christophe Philemotte.

A Gestalt description of permutation

A Gestalt description of permutation

For the TSP, each variable \(x_{i}\in X\) represents a city and a candidate solution describes a tour of cities or a permutation \(p=(\ldots,x_{i},\ldots). \) The permutation is defined by the mapping function \(\mathcal{D}\) that maps each city onto an integer \(1,\ldots,|X|\) such that each integer is mapped to an unique city x i .

With a lens, a gestalt description is defined. The variables, i.e. the cities, are then aggregated into gestalt variables G i , or megacities. The permutation is defined over the set of megacities and a candidate solution is reformulated as a tour of megacities p = (…, G i , …). In a megacity, the cities are also ordered. This order is determined while applying a new lens and respect the initial order and proximity of the cities. And the last city of a megacity is connected to the first city of the next megacity and so on. A direct bijection exists between a permutation of megacities and permutation of cities. Thus, the same operators can be simply applied by considering the new level of representation. For instance, the PMX will work by matching labels of megacities and ERX by recombining the edges between megacities.

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Philemotte, C., Bersini, H. The Gestalt heuristic: emerging abstraction to improve combinatorial search. Nat Comput 11, 499–517 (2012). https://doi.org/10.1007/s11047-012-9317-x

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