, Volume 73, Issue 3, pp 483–510 | Cite as

A Quantization Framework for Smoothed Analysis of Euclidean Optimization Problems



We consider the smoothed analysis of Euclidean optimization problems. Here, input points are sampled according to density functions that are bounded by a sufficiently small smoothness parameter \(\phi \). For such inputs, we provide a general and systematic approach that allows designing linear-time approximation algorithms whose output is asymptotically optimal, both in expectation and with high probability. Applications of our framework include maximum matching, maximum TSP, and the classical problems of k-means clustering and bin packing. Apart from generalizing corresponding average-case analyses, our results extend and simplify a polynomial-time probable approximation scheme on multidimensional bin packing on \(\phi \)-smooth instances, where \(\phi \) is constant (Karger and Onak in Polynomial approximation schemes for smoothed and random instances of multidimensional packing problems, pp 1207–1216, 2007). Both techniques and applications of our rounding-based approach are orthogonal to the only other framework for smoothed analysis of Euclidean problems we are aware of (Bläser et al. in Algorithmica 66(2):397–418, 2013).


Smoothed analysis Euclidean optimization problems  Bin packing Maximum matching Maximum traveling salesman problem 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Saarbrücken Graduate School of Computer ScienceSaarbrückenGermany
  2. 2.Department of Computer ScienceSaarland UniversitySaarbrückenGermany
  3. 3.Max Planck Institute for InformaticsSaarbrückenGermany

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