Abstract a recent proof of np hardness of euclidean sum ofsquares clustering, due to drineas et al. Nphardness of balanced minimum sumofsquares clustering. In this brief note, we will show that kmeans clustering is nphard even in d 2 dimensions. We convert, within polynomialtime and sequential processing, an npcomplete problem into a real. Agglomerative algorithms start with each point as a separate cluster and successively merge the most similar pair of clusters. But this bound seems to be particularly hard to compute. In the 2dimensional euclidean version of tsp problem, we are given a set of ncities in a plane and the pairwise distances between them. If you would take the sum of the last array it would be correct. Keywords clustering sumofsquares complexity 1 introduction clustering is a powerful tool for automated analysis of data.
If you have not read it yet, i recommend starting with part 1. How to calculate within group sum of squares for kmeans. This gap in understanding left open the intriguing possibility that the problem might admit a ptas for all k. This results in a partitioning of the data space into voronoi cells. See nphardness of euclidean sumofsquares clustering, aloise et. Approximation algorithms for nphard clustering problems ramgopal r. Mettu 103014 3 measuring cluster quality the cost of a set of cluster centers is the sum, over all points, of the weighted distance from each point to the. Clustering is one of the most popular data mining methods, not only due to its exploratory power, but also as a preprocessing step or subroutine for other techniques. Since the inverse square of a negative number is equal to the inverse square of the corresponding positive number, 3 is twice 2. A recent proof of np hardness of euclidean sumofsquares clustering, due to drineas et al. We present the algorithms and hardness results for clustering ats for many possible combinations of kand, where each of them either is the rst result or signi cantly improves the previous results for the given values for kand. Most quantization methods are essentially based on data clustering algorithms. For euclidean metric when the center could be any point in the space, the upper bound is still 2 and the best hardness of approximation is a factor 1. We show in this paper that this problem is nphard in general.
Brouwers xed point given a continuous function fmapping a compact convex set to itself, brouwers xed point theorem guarantees that fhas a xed point, i. In the kmeans clustering problem we are given a nite set of points sin rd, an integer k 1, and the goal is to nd kpoints usually called centers so to minimize the sum of the squared euclidean distance of each point in sto its closest center. Perhaps variations of the subset sum problem if some vertices have negative weights s. Eppstein, uc irvine, cs seminar spring 2006 quality guarantees for our approximation algorithm lower bound. Solving the minimum sumofsquares clustering problem by. Notice the kmeans clustering of multidimensional data is nphard 11. Other studies reported similar findings pertaining to the fuzzy cmeans algorithm. Clustering and sum of squares proofs, part 2 windows on. I got a little confused with the squares and the sums.
Keywords clustering sum ofsquares complexity 1 introduction clustering is a powerful tool for automated analysis of data. An agglomerative clustering method for large data sets. Popat, nphardness of euclidean sumofsquares clustering, machine learning, vol. Finally we can simplify 3 by multiplying each term by 4, obtaining x1 n1 1 n 122.
Euclidean sumofsquares clustering is an nphard problem, where we group n data points into k clusters. But its also unnecessarily complex because the offdiagonal elements are also calculated with np. Color quantization is an important operation with many applications in graphics and image processing. Clustering and sum of squares proofs, part 1 windows on. An agglomerative clustering method for large data sets omar kettani, faycal ramdani, benaissa tadili. Euclidean geometry is a mathematical system attributed to alexandrian greek mathematician euclid, which he described in his textbook on geometry. Nphardness of some quadratic euclidean 2clustering. You dont need to know the centroids coordinates the group means they pass invisibly on the background. Abstract a recent proof of nphardness of euclidean sumofsquares clustering, due to drineas et al. The minimum sumofsquares clustering mssc, also known in the literature as kmeans clustering, is a central problem in cluster analysis. A strongly nphard problem of partitioning a finite set of points of euclidean space into two clusters is considered. Cse 255 lecture 6 data mining and predictive analytics community detection.
In these problems, the following criteria are minimized. Daniel aloise, amit deshpande, pierre hansen, and preyas popat. Siam journal on scientific computing, 214, 14851505. The use of multiple measurements in taxonomic problems.
Nphardness of deciding convexity of quartic polynomials. As for the hardness of checking nonnegativity of biquadratic forms. Maximizing the sum of the squares of numbers whose sum is. Many decision treebased packet classification algorithms have been developed and adopted in the commercial equipment, but they build just a suboptimal decision trees due to the computational complexity. Instruction how you can compute sums of squares sst, ssb, ssw out of matrix of distances euclidean between cases data points without having at hand the cases x variables dataset. As for the hardness of checking nonnegativity of biquadratic forms, we know of two different proofs. Approximation algorithms for nphard clustering problems. Given a set of n points x x 1, x n in a given euclidean space r q, it addresses the problem of finding a partition p c 1, c k of k clusters minimizing the sum of squared distances from each point to the centroid of the cluster to which it belongs. Maximizing the sum of the squares of numbers whose sum is constant.
In particular, we were not able either to find a polynomialtime algorithm to compute this bound, or to prove that the problem is nphard. Best possible leveli clustering formed by removing 2i1 largest edges from mst therefore, summing over all levels in the clustering, with w. Intense recent discussions have focused on how to provide individuals with control over when their data can and cannot be used the eus right to. In general metrics, this problem admits a 2factor approximation which is also optimal assuming p6np18. In 3 we sum the inverse squares of all odd integers including the negative ones. The strong np hardness of problem 1 was proved in ageev et al. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.
The balanced clustering problem consists of partitioning a set of n objects into k equalsized clusters as long as n is a multiple of k. How does one prove the solution of minimum euclidean norm. A branchandcut sdpbased algorithm for minimum sumof. Hardness and algorithms euiwoong lee and leonard j. Colour quantisation using the adaptive distributing units. Also, if you find errors please mention them in the comments or otherwise get in touch with me and i will fix them asap welcome back. Abstract a recent proof of np hardness of euclidean sumofsquares clustering, due to drineas et al. Cambridge core knowledge management, databases and data mining data management for multimedia retrieval by k. This is the first post in a series which will appear on windows on theory in the coming weeks.
The strong nphardness of problem 1 was proved in ageev et al. Although many of euclids results had been stated by earlier mathematicians, euclid was the first to show. Problem 7 minimum sum of normalized squares of norms clustering. Euclids method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. In this respect, a sufficient condition for the problem to be nphard, is. I am excited to temporarily join the windows on theory family as a guest blogger. Np hardness of some quadratic euclidean 2 clustering problems.
I have a list of 100 values in python where each value in the list corresponds to an ndimensional list. The hardness of approximation of euclidean kmeans authors. Schulmany department of computer science california institute of technology july 3, 2012 abstract we study a generalization of the famous kcenter problem where each object is an a ne subspace of dimension, and give either the rst or signi cantly improved algorithms and. The main di culty in obtaining hardness results stems from the euclidean nature of the problem, and the fact that any point in rd can be a potential center. Minimum sumofsquares clustering mssc consists in partitioning a given set of n points into k clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. Proving nphardness of strange graph partition problem. An interior point algorithm for minimum sumofsquares clustering. Pdf nphardness of some quadratic euclidean 2clustering.
Pranjal awasthi, moses charikar, ravishankar krishnaswamy, ali kemal sinop submitted on 11 feb 2015. The nphardness of checking nonnegativity of quartic forms follows, e. A recent proof of nphardness of euclidean sumofsquares clustering, due to drineas et al. Nphardness of optimizing the sum of rational linear. I have data set with 318 data points and 11 attributes.
Let us consider two problems, the traveling salesperson tsp and the clique, as illustration. This is part 2 of a series on clustering, gaussian mixtures, and sum of squares sos proofs. A popular clustering criterion when the objects are points of a qdimensional space is the minimum sum of squared distances from each point to the centroid of the cluster to which it belongs. Recent studies have demonstrated the effectiveness of hard cmeans kmeans clustering algorithm in this domain. Jonathan alon, stan sclaroff, george kollios, and vladimir pavlovic. A key contribution of this work is a dynamic programming dp based algorithm with on 2 k complexity, which produces the. The solution criterion is the minimum of the sum over both clusters of. Np hardness of partitioning a graph into two subgraphs. Hard versus fuzzy cmeans clustering for color quantization. I am trying to find the best number of cluster required for my data set. There are two main strategies for solving clustering problems.