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On the Minimal Constraint Satisfaction Problem: Complexity and Generation

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Combinatorial Optimization and Applications

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9486))

Abstract

The Minimal Constraint Satisfaction Problem, or Minimal CSP for short, arises in a number of real-world applications, most notably in constraint-based product configuration. Despite its very permissive structure, it is NP-hard, even when bounding the size of the domains by \(d\ge 9\). Yet very little is known about the Minimal CSP beyond that. Our contribution through this paper is twofold. Firstly, we generalize the complexity result to any value of d. We prove that the Minimal CSP remains NP-hard for \(d\ge 3\), as well as for \(d=2\) if the arity k of the instances is strictly greater than 2. Our complexity result can be seen as providing a dichotomy theorem for the Minimal CSP. Secondly, we build a generator that can create Minimal CSP instances of any size, using the constrainedness as a parameter. Our generator can be used to study behaviors that are typical of NP-hard problems, such as the presence of a phase transition, in the case of the Minimal CSP.

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Notes

  1. 1.

    We are aware that finding a solution to a Minimal CSP is both a search problem (find a solution) and a promise problem (the input CSP is satisfiable because it is minimal), rather that a decision problem. However, we use this terminology in the same manner as Gottlob [6], where a thorough discussion of the matter can be found.

  2. 2.

    Our experiments are run under CentOS 6.6, on two Intel processors (1.33 Ghz each), and with 12 GB of DDR2 FB-DIMM RAM.

References

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This publication has emanated from research conducted with the financial support of Science Foundation Ireland (SFI) under Grant Number SFI/12/RC/2289.

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Authors and Affiliations

  1. Department of Computer Science, Insight Centre for Data Analytics, University College Cork, Cork, Ireland

    Guillaume Escamocher & Barry O’Sullivan

Authors
  1. Guillaume Escamocher
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  2. Barry O’Sullivan
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Corresponding author

Correspondence to Guillaume Escamocher .

Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Marywood University, Scranton, Pennsylvania, USA

    Zaixin Lu

  2. North Carolina Central University, Durham, North Carolina, USA

    Donghyun Kim

  3. University of Texas at Dallas, Richardson, Texas, USA

    Weili Wu

  4. Texas Southern University, Houston, Texas, USA

    Wei Li

  5. University of Texas at Dallas, Richardson, Texas, USA

    Ding-Zhu Du

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Escamocher, G., O’Sullivan, B. (2015). On the Minimal Constraint Satisfaction Problem: Complexity and Generation. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_54

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  • DOI: https://doi.org/10.1007/978-3-319-26626-8_54

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26625-1

  • Online ISBN: 978-3-319-26626-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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