W. Fernandez de la Vega
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.
- N. Alon, W. Fernandez de la Vega, R. Kannan and M. Karpinski, Random Sampling and MAX-CSP Problems, Proc. 34th ACM STOC (2002), pp. 232--239. Google Scholar
Digital Library
- S. Arora, D. Karger and M. Karpinski, Polynomial Time Approximation Schemes for Dense Instances of NP-Hard Problems, Proc. 27th STOC (1995), pp. 284-293; J. Computer and System Sciences 58 (1999), pp. 193--210. Google ScholarDigital Library
- W. Fernandez de la Vega, MAX-CUT has a Randomized Approximation Scheme in Dense Graphs, Random Structures and Algorithms 8 (1996), pp. 187--198. Google ScholarDigital Library
- W. Fernandez de la Vega and M. Karpinski, Polynomial time approximation of dense weighted instances of MAX-CUT, Random Structures and Algorithms 16 (2000), pp. 314--332. Google ScholarDigital Library
- W. Fernandez de la Vega and C. Kenyon, A randomized approximation scheme for metric MAX-CUT, Proc. 39th IEEE FOCS (1998), pp. 468--471, final version in J. Computer and System Sciences 63 (2001), pp. 531--541. Google ScholarDigital Library
- W. Fernandez de la Vega, M. Karpinski, C. Kenyon and Y. Rabani, Approximation schemes for clustering problems, Proc. 35th ACM STOC (2003), pp. 50--58. Google ScholarDigital Library
- W. Fernandez de la Vega, M. Karpinski and C. Kenyon, Approximation Schemes for Metric Bisection and Partitioning, Proc. 15th ACM-SIAM SODA (2004), pp. 499--508. Google ScholarDigital Library
- A. M. Frieze and R. Kannan, The Regularity Lemma and Approximation Schemes for Dense Problems, Proc. 37th IEEE FOCS (1996), pp. 12--20. Google ScholarDigital Library
- A. M. Frieze and R. Kannan, Quick Approximation to Matrices and Applications, Combinatorica 19 (2) (1999), pp. 175--120.Google ScholarCross Ref
- A. M. Frieze, R. Kannan and S. Vempala, Fast Monte-Carlo Algorithms for Finding Low-Rank Approximations, J. of the ACM 51(6) (2004), pp. 1025--1041. Google ScholarDigital Library
- O. Goldreich, S. Goldwasser and D. Ron, Property Testing and its Connection to Learning and Approximation, Proc. 37th IEEE FOCS (1996), pp. 339--348; J. ACM 45 (1998), pp. 653--750. Google ScholarDigital Library
- P. Indyk, A Sublinear Time Approximation Scheme for Clustering in Metric Spaces, Proc. 40th IEEE FOCS (1999), pp. 154--159. Google ScholarDigital Library
- R. Macias and C. Segovia, Lipschitz functions on spaces of homogenous type, Advances in Mathematics 33 (1979), pp. 257--270.Google ScholarCross Ref
- F. McSherry, Spectral Partitioning of Random Graphs, FOCS 2001, pp.529--537 Google ScholarDigital Library
- R. R. Mettu and C. G. Plaxton, The Online Median Problem, Proc. 41st IEEE FOCS (2000), pp. 339--348. Google ScholarDigital Library
- S. Semmes, A brief introduction to Gromov's notion of hyperbolic groups, Mathematics, arXiv: math CA/021341 (2003), pp. 1--10.Google Scholar
- P. H. Sneath and R. R. Sokal, Numerical Taxonomy, Freeman, London, 1973.Google Scholar
Index Terms
- Tensor decomposition and approximation schemes for constraint satisfaction problems
Recommendations
Tensor Completion via Fully-Connected Tensor Network Decomposition with Regularized Factors
AbstractThe recently proposed fully-connected tensor network (FCTN) decomposition has a powerful ability to capture the low-rankness of tensors and has achieved great success in tensor completion. However, the FCTN decomposition-based method is highly ...
Ruling out polynomial-time approximation schemes for hard constraint satisfaction problems
CSR'07: Proceedings of the Second international conference on Computer Science: theory and applicationsThe maximum constraint satisfaction problem (Max CSP) is the following computational problem: an instance is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of ...
Alternating Linear Scheme in a Bayesian Framework for Low-Rank Tensor Approximation
Multiway data often naturally occurs in a tensorial format which can be approximately represented by a low-rank tensor decomposition. This is useful because complexity can be significantly reduced and the treatment of large-scale data sets can be ...
Comments