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Portfolio Approaches for Constraint Optimization Problems

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Learning and Intelligent Optimization (LION 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8426))

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Abstract

Within the Constraints Satisfaction Problems (CSP) context, a methodology that has proven to be particularly performant consists in using a portfolio of different constraint solvers. Nevertheless, comparatively few studies and investigations have been done in the world of Constraint Optimization Problems (COP). In this work, we provide a generalization to COP as well as an empirical evaluation of different state of the art existing CSP portfolio approaches properly adapted to deal with COP. Experimental results confirm the effectiveness of portfolios even in the optimization field, and could give rise to some interesting future research.

Work partially supported by Aeolus project, ANR-2010-SEGI-013-01.

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Notes

  1. 1.

    Formally, \(\mathtt{MIN }(i) = \min V_i\) and \(\mathtt{MAX }(i) = \max V_i\) where \(V_i = \{ \mathtt{val }(s, i, T) \ . \ s \in U \}\). Note that a portfolio solver executing more than one solver for \(t < T\) seconds could produce a solution that is worse than \(\mathtt{MIN }(i)\). This however is very uncommon: in our experiments we noticed that the 0 score was assigned only to the solvers that did not find any solution.

  2. 2.

    FlatZinc [6] is the low level language that each solver uses for solving a given MiniZinc instance. A key feature of FlatZinc is that, starting from a general MiniZinc model, every solver can produce a specialized FlatZinc by redefining the global constraints definitions. We noticed that, especially for huge instances, the time needed for extracting features was strongly dominated by the FlatZinc conversion. However, for the instances of the final dataset this time was in average 10.36 s, with a maximum value of 504 s and a median value of 3.17 s.

  3. 3.

    Following [1] methodology, CPX won all the elections we simulated using different criteria, viz.: Borda, Approval, and Plurality.

  4. 4.

    The objective function of the best approach considered was obtained by replacing that of the IP problem defined in [18] (we use the very same notation) with:

    $$\begin{aligned} \max \left[ C_1 \sum _y y_i + C_2 \sum _{i, S, t} \mathtt{score }(S, i, t) \cdot x_{S, t} + C_3 \sum _{S,t} t \cdot x_{S, t} \right] \end{aligned}$$

    where \(C_1 = -C^2\), \(C_2 = C\), \(C_3 = -\frac{1}{C}\), and adding the constraint \(\sum _t x_{S,t} \le 1\), \(\forall S\).

  5. 5.

    For more details, we defer the interested reader to [37].

  6. 6.

    To conduct the experiments we used Intel Dual-Core 2.93 GHz computers with 3 MB of CPU cache, 2 GB of RAM, and Ubuntu 12.04 operating system. For keeping track of the solving times we considered the CPU time by exploiting Unix time command.

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Authors and Affiliations

  1. Department of Computer Science and Engineering/Lab Focus INRIA, University of Bologna, Bologna, Italy

    Roberto Amadini, Maurizio Gabbrielli & Jacopo Mauro

Authors
  1. Roberto Amadini
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  2. Maurizio Gabbrielli
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  3. Jacopo Mauro
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Correspondence to Jacopo Mauro .

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Editors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. AT&T Labs Research, Middletown, New Jersey, USA

    Mauricio G.C. Resende

  3. University of Florida, Gainesville, Florida, USA

    Chrysafis Vogiatzis

  4. Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Jose L. Walteros

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Amadini, R., Gabbrielli, M., Mauro, J. (2014). Portfolio Approaches for Constraint Optimization Problems. In: Pardalos, P., Resende, M., Vogiatzis, C., Walteros, J. (eds) Learning and Intelligent Optimization. LION 2014. Lecture Notes in Computer Science(), vol 8426. Springer, Cham. https://doi.org/10.1007/978-3-319-09584-4_3

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  • DOI: https://doi.org/10.1007/978-3-319-09584-4_3

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