Local Search Yields Approximation Schemes for k-Means and k-Median in Euclidean and Minor-Free Metrics

We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs, (2) k-median and k-means in edge-weighted planar graphs, (3) k-means in Euclidean space of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the pth power of the shortest-path distance. The algorithm is local search where the local neighborhood of a solution S consists of all solutions obtained from S by removing and adding 1/εO(1) centers.

[1]  Timothy M. Chan,et al.  Approximation Algorithms for Maximum Independent Set of Pseudo-Disks , 2009, Discrete & Computational Geometry.

[2]  David M. Mount,et al.  A local search approximation algorithm for k-means clustering , 2002, SCG '02.

[3]  Shi Li,et al.  Approximating k-median via pseudo-approximation , 2012, STOC '13.

[4]  Venkatesan Guruswami,et al.  Embeddings and non-approximability of geometric problems , 2003, SODA '03.

[5]  Maria-Florina Balcan,et al.  Approximate clustering without the approximation , 2009, SODA.

[6]  Satish Rao,et al.  Approximation schemes for Euclidean k-medians and related problems , 1998, STOC '98.

[7]  Sergei Vassilvitskii,et al.  Worst-case and Smoothed Analysis of the ICP Algorithm, with an Application to the k-means Method , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[8]  Piotr Indyk,et al.  Approximate clustering via core-sets , 2002, STOC '02.

[9]  Sudipto Guha,et al.  Clustering Data Streams: Theory and Practice , 2003, IEEE Trans. Knowl. Data Eng..

[10]  Amit Kumar,et al.  Linear-time approximation schemes for clustering problems in any dimensions , 2010, JACM.

[11]  Sariel Har-Peled,et al.  Smaller Coresets for k-Median and k-Means Clustering , 2005, SCG.

[12]  Kent Quanrud,et al.  Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs , 2015, ESA.

[13]  Yves Crama,et al.  Local Search in Combinatorial Optimization , 2018, Artificial Neural Networks.

[14]  Aditya Bhaskara,et al.  Distributed Balanced Clustering via Mapping Coresets , 2014, NIPS.

[15]  Bodo Manthey,et al.  Smoothed Analysis of the k-Means Method , 2011, JACM.

[16]  Guy E. Blelloch,et al.  Parallel approximation algorithms for facility-location problems , 2010, SPAA '10.

[17]  Shi Li,et al.  A 1.488 approximation algorithm for the uncapacitated facility location problem , 2011, Inf. Comput..

[18]  Sariel Har-Peled,et al.  On coresets for k-means and k-median clustering , 2004, STOC '04.

[19]  Rajmohan Rajaraman,et al.  Analysis of a local search heuristic for facility location problems , 2000, SODA '98.

[20]  Amit Kumar,et al.  A simple linear time (1 + /spl epsiv/)-approximation algorithm for k-means clustering in any dimensions , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[21]  Nathan Linial,et al.  Are Stable Instances Easy? , 2009, Combinatorics, Probability and Computing.

[22]  Michael Langberg,et al.  A unified framework for approximating and clustering data , 2011, STOC '11.

[23]  Sariel Har-Peled,et al.  Separating a Voronoi Diagram via Local Search , 2013, SoCG.

[24]  Dorit S. Hochbaum,et al.  Heuristics for the fixed cost median problem , 1982, Math. Program..

[25]  Philip N. Klein,et al.  The power of local search for clustering , 2016, ArXiv.

[26]  Claire Mathieu,et al.  Effectiveness of Local Search for Geometric Optimization , 2015, SoCG.

[27]  Sayan Bandyapadhyay,et al.  On Variants of k-means Clustering , 2015, SoCG.

[28]  Mohammad R. Salavatipour,et al.  Local Search Yields a PTAS for k-Means in Doubling Metrics , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[29]  Avrim Blum,et al.  Stability Yields a PTAS for k-Median and k-Means Clustering , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[30]  Jiawei Zhang,et al.  Approximation algorithms for facility location problems , 2004 .

[31]  Pranjal Awasthi,et al.  Improved Spectral-Norm Bounds for Clustering , 2012, APPROX-RANDOM.

[32]  Ravishankar Krishnaswamy,et al.  The Hardness of Approximation of Euclidean k-Means , 2015, SoCG.

[33]  Satish Rao,et al.  A Nearly Linear-Time Approximation Scheme for the Euclidean k-Median Problem , 2007, SIAM J. Comput..

[34]  Samir Khuller,et al.  Greedy strikes back: improved facility location algorithms , 1998, SODA '98.

[35]  Sudipto Guha,et al.  Improved Combinatorial Algorithms for Facility Location Problems , 2005, SIAM J. Comput..

[36]  Kamesh Munagala,et al.  Local Search Heuristics for k-Median and Facility Location Problems , 2004, SIAM J. Comput..

[37]  Vijay V. Vazirani,et al.  Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation , 2001, JACM.

[38]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[39]  Mary Inaba,et al.  Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering: (extended abstract) , 1994, SCG '94.

[40]  Maria-Florina Balcan,et al.  Clustering under Perturbation Resilience , 2011, SIAM J. Comput..

[41]  Amin Saberi,et al.  A new greedy approach for facility location problems , 2002, STOC '02.

[42]  Dan Feldman,et al.  A PTAS for k-means clustering based on weak coresets , 2007, SCG '07.

[43]  Robin Thomas,et al.  A separator theorem for graphs with an excluded minor and its applications , 1990, STOC '90.

[44]  Amit Kumar,et al.  Clustering with Spectral Norm and the k-Means Algorithm , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[45]  David B. Shmoys,et al.  Approximation algorithms for facility location problems , 2000, APPROX.

[46]  Greg N. Frederickson,et al.  Fast Algorithms for Shortest Paths in Planar Graphs, with Applications , 1987, SIAM J. Comput..