Variable Ordering for the Application of BDDs to the Maximum Independent Set Problem

  • David Bergman
  • Andre A. Cire
  • Willem-Jan van Hoeve
  • John N. Hooker
Conference paper

DOI: 10.1007/978-3-642-29828-8_3

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7298)
Cite this paper as:
Bergman D., Cire A.A., van Hoeve WJ., Hooker J.N. (2012) Variable Ordering for the Application of BDDs to the Maximum Independent Set Problem. In: Beldiceanu N., Jussien N., Pinson É. (eds) Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems. CPAIOR 2012. Lecture Notes in Computer Science, vol 7298. Springer, Berlin, Heidelberg

Abstract

The ordering of variables can have a significant effect on the size of the reduced binary decision diagram (BDD) that represents the set of solutions to a combinatorial optimization problem. It also influences the quality of the objective function bound provided by a limited-width relaxation of the BDD. We investigate these effects for the maximum independent set problem. By identifying variable orderings for the BDD, we show that the width of an exact BDD can be given a theoretical upper bound for certain classes of graphs. In addition, we draw an interesting connection between the Fibonacci numbers and the width of exact BDDs for general graphs. We propose variable ordering heuristics inspired by these results, as well as a k-layer look-ahead heuristic applicable to any problem domain. We find experimentally that orderings that result in smaller exact BDDs have a strong tendency to produce tighter bounds in relaxation BDDs.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • David Bergman
    • 1
  • Andre A. Cire
    • 1
  • Willem-Jan van Hoeve
    • 1
  • John N. Hooker
    • 1
  1. 1.Tepper School of BusinessCarnegie Mellon UniversityPittsburghU.S.A.

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