Finding independent sets in a graph using continuous multivariable polynomial formulations

  • Authors:

  • James Abello;Sergiy Butenko;Panos M. Pardalos;Mauricio G. C. Resende

  • Affiliations:

  • Shannon Laboratory, AT&T Labs Research, Florham Park, NJ 07932 USA. (e-mails: abello@research.att.com);Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611 USA. (e-mails: butenko@ufl.edu);Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611 USA. (e-mails: pardalos@ufl.edu);Shannon Laboratory, AT&T Labs Research, Florham Park, NJ 07932 USA. (e-mails: mgcr@research.att.com)

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2001

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Abstract

Two continuous formulations of the maximum independent set problem on a graph G=(V,E) are considered. Both cases involve the maximization of an n-variable polynomial over the n-dimensional hypercube, where n is the number of nodes in G. Two (polynomial) objective functions F(x) and H(x) are considered. Given any solution to x0 in the hypercube, we propose two polynomial-time algorithms based on these formulations, for finding maximal independent sets with cardinality greater than or equal to F(x0) and H(x0), respectively. A relation between the two approaches is studied and a more general statement for dominating sets is proved. Results of preliminary computational experiments for some of the DIMACS clique benchmark graphs are presented.