Tensor decomposition and approximation schemes for constraint satisfaction problems

Authors:
W. Fernandez de la Vega;Marek Karpinski;Ravi Kannan;Santosh Vempala
Affiliations:
LRI, Université de Paris-Sud, Orsay, France;University of Bonn, Bonn;Yale, New Haven, CT;MIT, Cambridge, MA
Venue:
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Year:
2005

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Abstract

The only general class of MAX-rCSP problems for which Polynomial Time Approximation Schemes (PTAS) are known are the dense problems. In this paper, we give PTAS's for a much larger class of weighted MAX-rCSP problems which includes as special cases the dense problems and, for r = 2, all metric instances (where the weights satisfy the triangle inequality) and quasimetric instances; for r 2, our class includes a generalization of metrics. Our algorithms are based on low-rank approximations with two novel features: (1) a method of approximating a tensor by the sum of a small number of "rank-1" tensors, akin to the traditional Singular Value Decomposition (this might be of independent interest) and (2) a simple way of scaling the weights. Besides MAX-rCSP problems, we also give PTAS's for problems with a constant number of global constraints such as maximum weighted graph bisection and some generalizations.