# Clustering with or without the approximation

## Abstract

We study algorithms for clustering data that were recently proposed by Balcan et al. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) and that have already given rise to several follow-up papers. The input for the clustering problem consists of points in a metric space and a number *k*, specifying the desired number of clusters. The algorithms find a clustering that is provably close to a target clustering, provided that the instance has the “(1+*α*,*ε*)-property”, which means that the instance is such that all solutions to the *k*-median problem for which the objective value is at most (1+*α*) times the optimal objective value correspond to clusterings that misclassify at most an *ε* fraction of the points with respect to the target clustering. We investigate the theoretical and practical implications of their results.

Our main contributions are as follows. First, we show that instances that have the (1+*α*,*ε*)-property and for which, additionally, the clusters in the target clustering are large, are easier than general instances: the algorithm proposed in Balcan et al. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) is a constant factor approximation algorithm with an approximation guarantee that is better than the known hardness of approximation for general instances. Further, we show that it is *NP*-hard to check if an instance satisfies the (1+*α*,*ε*)-property for a given (*α*,*ε*); the algorithms in Balcan et al. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) need such *α* and *ε* as input parameters, however. We propose ways to use their algorithms even if we do not know values of *α* and *ε* for which the assumption holds. Finally, we implement these methods and other popular methods, and test them on real world data sets. We find that on these data sets there are no *α* and *ε* so that the dataset has both (1+*α*,*ε*)-property and sufficiently large clusters in the target solution. For the general case where there are no assumptions about the cluster sizes, we show that on our data sets the performance guarantee proved by Balcan et a. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) is meaningless for the values of *α*,*ε* for which the data set has the (1+*α*,*ε*)-property. The algorithm nonetheless gives reasonable results, although it is outperformed by other methods.

## Keywords

Clustering*k*-median Algorithms Approximation

## Preview

Unable to display preview. Download preview PDF.

## References

- Arora S, Raghavan P, Rao S (1999) Approximation schemes for Euclidean
*k*-medians and related problems. In: STOC ’98: proceedings of the 30th annual ACM symposium on theory of computing, pp 106–113 Google Scholar - Arthur D, Vassilvitskii S (2006) How slow is the k-means method? In: SCG ’06: 22d annual symposium on computational geometry, pp 144–153 CrossRefGoogle Scholar
- Arthur D, Vassilvitskii S (2007) k-means++: the advantages of careful seeding. In: SODA ’07: 18th annual ACM-SIAM symposium on discrete algorithms, pp 1027–1035 Google Scholar
- Arya V, Garg N, Khandekar R, Meyerson A, Munagala K, Pandit V (2004) Local search heuristics for
*k*-median and facility location problems. SIAM J Comput 33(3):544–562 MathSciNetCrossRefzbMATHGoogle Scholar - Asuncion A, Newman D (2007) UCI machine learning repository. http://www.ics.uci.edu/mlearn/MLRepository.html
- Awasthi P, Blum A, Sheffet O (2010) Clustering under natural stability assumptions. http://repository.cmu.edu/compsci/123/, retrieved on June 9th, 2010
- Balcan MF, Braverman M (2009) Finding low error clusterings. In: COLT 2009: 22nd annual conference on learning theory Google Scholar
- Balcan MF, Blum A, Vempala S (2008) A discriminative framework for clustering via similarity functions. In: STOC 2008: 40th annual ACM symposium on theory of computing, pp 671–680 Google Scholar
- Balcan MF, Blum A, Gupta A (2009a) Approximate clustering without the approximation. In: SODA ’09: 19th annual ACM-SIAM symposium on discrete algorithms, pp 1068–1077 Google Scholar
- Balcan MF, Röglin H, Teng SH (2009b) Agnostic clustering. In: ALT 2009: 20th international conference on algorithmic learning theory. Lecture notes in computer science, vol 5809. Springer, Berlin, pp 384–398 Google Scholar
- Balcan MF, Röglin H, Teng S, Voevodski K, Xia Y (2010) Efficient clustering with limited distance information. In: UAI 2010: the 26th conference on uncertainty in artificial intelligence Google Scholar
- Beasley JE (1985a) A note on solving large p-median problems. Eur J Oper Res 21(2):270–273 MathSciNetCrossRefzbMATHGoogle Scholar
- Beasley JE (1985b) OR-Library
*p*-median—uncapacitated. http://people.brunel.ac.uk/mastjjb/jeb/orlib/pmedinfo.html - Bilu Y, Linial N (2010) Are stable instances easy? In: ICS 2010: the first symposium on innovations in computer science, pp 332–341 Google Scholar
- Charikar M, Guha S (2005) Improved combinatorial algorithms for facility location problems. SIAM J Comput 34(4):803–824 (electronic) MathSciNetCrossRefzbMATHGoogle Scholar
- Charikar M, Guha S, Tardos É, Shmoys DB (2002) A constant-factor approximation algorithm for the
*k*-median problem. J Comput Syst Sci 65(1):129–149 MathSciNetCrossRefzbMATHGoogle Scholar - Feige U (1998) A threshold of ln
*n*for approximating set cover. J ACM 45(4):634–652 MathSciNetCrossRefzbMATHGoogle Scholar - Gupta A (2009) personal communication Google Scholar
- Jain K, Vazirani VV (2001) Approximation algorithms for metric facility location and
*k*-median problems using the primal-dual schema and Lagrangian relaxation. J ACM 48(2):274–296 MathSciNetCrossRefzbMATHGoogle Scholar - Jain K, Mahdian M, Markakis E, Saberi A, Vazirani VV (2003) Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J ACM 50(6):795–824 MathSciNetCrossRefGoogle Scholar
- Lloyd S (1982) Least squares quantization in PCM. IEEE Trans Inf Theory 28(2):129–137 MathSciNetCrossRefzbMATHGoogle Scholar
- Ostrovsky R, Rabani Y, Schulman LJ, Swamy C (2006) The effectiveness of Lloyd-type methods for the k-means problem. In: FOCS ’06: 47th annual IEEE symposium on foundations of computer science, pp 165–176 Google Scholar