Random sampling and approximation of MAX-CSP problems

  • Authors:

  • Noga Alon;W. Fernandez de la Vega;Ravi Kannan;Marek Karpinski

  • Affiliations:

  • Tel Aviv University, Tel Aviv and IAS, Princeton;CNRS, Universite Paris Sud, Orsay;Yale University;University of Bonn

  • Venue:
  • STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
  • Year:
  • 2002

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Abstract

We present a new efficient sampling method for approximating r-dimensional Maximum Constraint Satisfaction Problems, MAX-rCSP, on n variables up to an additive error &egr;nr. We prove a newgeneral paradigm in that it suffices, for a given set of constraints, to pick a small uniformly random subset of its variables, and the optimum value of the subsystem induced on these variables gives (after a direct normalization and with high probability) an approximation to the optimum of the whole system up to an additive error of &egr;nr. Our method gives for the first time a polynomial in &egr;—1 bound on the sample size necessary to carry out the above approximation. Moreover, this bound is independent in the exponent on the dimension r. The above method gives a completely uniform sampling technique for all the MAX-rCSP problems, and improves the best known sample bounds for the low dimensional problems, like MAX-CUT. The method of solution depends on a new result on t he cut norm of random subarrays, and a new sampling technique for high dimensional linear programs. This method could be also of independent interest.