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Epiphytic Trees: Relational Consistency Applied to Global Optimization Problems

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Integration of Constraint Programming, Artificial Intelligence, and Operations Research (CPAIOR 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10848))

Abstract

Much effort has been spent to identify classes of CSPs in terms of the relationship between network structure and the amount of consistency that guarantees a backtrack-free solution. In this paper, we address Numerical Constrained global Optimization Problems (NCOPs) encoded as ternary networks, characterizing a class of such problems for which a combination of Generalized Arc-Consistency (GAC) and Relational Arc-Consistency (RAC) is sufficient to ensure a backtrack-free solution, called Epiphytic Trees. While GAC is a domain filtering technique, enforcing RAC creates new constraints in the network. Alternatively, we propose a branch and bound method to achieve a relaxed form of RAC, thus finding an approximation of the solution of NCOPs. We empirically show that Epiphytic Trees are relevant in practice. In addition, we extend this class to cover all ternary NCOPs, for which Strong Directional Relational k-Consistency ensures a backtrack-free solution.

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Notes

  1. 1.

    As occur with the network \(x_1=x_2\) and \(x_1=0.5 \cdot x_2\), where only the box \(B = ([0,0], [0,0])\) turns both the constraints GAC but the contractor does an endless bisection on any initial arbitrary box.

  2. 2.

    Obtained by encoding NCOPs.

  3. 3.

    The same problem occurs with the results of Freuder [21] and Jégou [26]: achieving (hyper-)k-consistency creates new constraints of arity \(k-1\) thus changing the width of the (hyper)graph.

  4. 4.

    Given \(B, B' \in \mathbb {I}^n\) and an interval function \(\mathcal {F}\), if \(B' \subseteq B\), then \(\mathcal {F}(B') \subseteq \mathcal {F}(B)\).

  5. 5.

    Using an AC-3 [31] style algorithm with an iteration counter to avoid looping.

  6. 6.

    A variant of 3-relational consistency where the consistency is considered with relation to two variables instead of one (see [15]).

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Acknowledgments

This work was supported by CAPES - Brazilian Federal Agency for Support and Evaluation of Graduate Education within the Ministry of Education of Brazil.

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  1. Federal University of Paraná, Curitiba, PR, Brazil

    Guilherme Alex Derenievicz & Fabiano Silva

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  1. Guilherme Alex Derenievicz
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Correspondence to Guilherme Alex Derenievicz .

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  1. Carnegie Mellon University, Pittsburgh, Pennsylvania, USA

    Willem-Jan van Hoeve

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Derenievicz, G.A., Silva, F. (2018). Epiphytic Trees: Relational Consistency Applied to Global Optimization Problems. In: van Hoeve, WJ. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2018. Lecture Notes in Computer Science(), vol 10848. Springer, Cham. https://doi.org/10.1007/978-3-319-93031-2_11

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  • DOI: https://doi.org/10.1007/978-3-319-93031-2_11

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