Keywords clustering sum ofsquares complexity 1 introduction clustering is a powerful tool for automated analysis of data. 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. Nphardness of euclidean sumofsquares clustering springerlink. The np hardness of checking nonnegativity of quartic forms follows, e. Given a set of observations x 1, x 2, x n, where each observation is a ddimensional real vector, kmeans clustering aims to partition the n observations into k sets k. I have a list of 100 values in python where each value in the list corresponds to an ndimensional list. The most celebrated example is, of course, the theory of np hardness. A branchandcut sdpbased algorithm for minimum sumof. An interior point algorithm for minimum sum of squares clustering. The benefit of kmedoid is it is more robust, because it minimizes a sum of dissimilarities instead of a sum of squared euclidean distances. In this respect, a sufficient condition for the problem to be nphard, is. I am trying to find the best number of cluster required for my data set. Approximation algorithms for nphard clustering problems ramgopal r. In this brief note, we will show that kmeans clustering is np hard even in d 2 dimensions.
Our best lower bounds are in bounded depth models or in monotone models, or use techniques such as diagonalization which usually apply only to somewhat contrived problems. 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. Let us consider two problems, the traveling salesperson tsp and the clique, as illustration. In the 2dimensional euclidean version of tsp problem, we are given a set of ncities in a plane and the pairwise distances between them.
Nphard in general euclidean space of d dimensions even for two clusters. Quadratic euclidean 1mean and 1median 2clustering problem. The solution criterion is the minimum of the sum over both clusters of. Nphardness of euclidean sumofsquares clustering machine. Strict monotonicity of sum of squares error and normalized. If you have not read it yet, i recommend starting with part 1.
I got a little confused with the squares and the sums. The hardness of approximation of euclidean kmeans authors. After that, with a sum of squares proof in hand, we will finish designing our mixture of gaussians algorithm for the onedimensional case. The minimum sumofsquares clustering mssc, also known in the literature as kmeans clustering, is a central problem in cluster analysis. Introduction dependence and distribution toward an extension to the multivariate case on clustering. A survey on exact methods for minimum sumofsquares clustering pierre hansen1, and daniel aloise2 1 gerad and hec montr. However, in reality, data objects often do not come fully equipped with a mapping into euclidean. Our work resolves an open question of kannan vempala, now, 2009. The architectural organization of a mobile radio network via a distributed algorithm. I have data set with 318 data points and 11 attributes. Minimum sum of squares 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. Abstract a recent proof of np hardness of euclidean sum ofsquares clustering, due to drineas et al. A large number of basic combinatorial problems are as hard as any of the problems in np. To formulate the original clustering problem as a min.
A survey on exact methods for minimum sumofsquares clustering. A randomized constantfactor approximation algorithm for the kmedian problem that runs in. Taking the sum of sqares for this matrix should work like. Nphardness of optimizing the sum of rational linear. Euclidean space, clustering, 2partition, quadratic.
There are prerequisites associated with clustering in vanets. 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. How to calculate within group sum of squares for kmeans. 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. Euclidean space into two clusters minimizing the sum of the squared distances. Color quantization is an important operation with many applications in graphics and image processing. Thesisnphardness of euclidean sumofsquares clustering. This is part 2 of a series on clustering, gaussian mixtures, and sum of squares sos proofs. The strong nphardness of problem 1 was proved in ageev et al. In this problem the criterion is minimizing the sum over all clusters of norms of the sum of cluster elements. Jan 24, 2009 a recent proof of nphardness of euclidean sumofsquares clustering, due to drineas et al. Np hardness of euclidean sumofsquares clustering article pdf available in machine learning 752. Other studies reported similar findings pertaining to the fuzzy cmeans algorithm. Pdf nphardness of some quadratic euclidean 2clustering.
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. The term kmeans was first used by james macqueen in 1967, 1. Hardness of checking maximal magnitude of a sum of a subset of vectors. In this brief note, we will show that kmeans clustering is nphard even in d 2 dimensions. Abstract a recent proof of nphardness of euclidean sumofsquares clustering, due to drineas et al. A survey on exact methods for minimum sumofsquares. The minimum sumofsquares clustering mssc formulation produces a mathematical problem of global optimization. Approximation algorithms for nphard clustering problems. Nphardness of deciding convexity of quartic polynomials. How to prove the nphardness or npcompleteness of this assignment problem.
Nphardness of deciding convexity of quartic polynomials and related problems. Siam journal on scientific computing, 214, 14851505. During my thesis i came across the following optimi. Nphardness of some quadratic euclidean 2clustering problems. Problem 7 minimum sum of normalized squares of norms clustering. Clustering and sum of squares proofs, part 1 windows on theory. Here ive written out the squared euclidean distance as a quadratic. We prove that the problem is strongly nphard and there is no fully polynomial time approximation scheme for its solution. We prove the strong nphardness for problem 1 with general loss functions.
Outline 1 introduction clustering minimum sumofsquares clustering computational complexity kmeans. Clustering is a scientific method which addresses the. Hard versus fuzzy cmeans clustering for color quantization. But this bound seems to be particularly hard to compute. Nphardness of euclidean sumofsquares clustering article pdf available in machine learning 752. Nphardness of balanced minimum sumofsquares clustering. Strong nphardness for sparse optimization with concave.
The reason we need complexity assumptions like the 3sum conjecture is that we cannot prove meaningful lower bounds in any nontrivial computational models. 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. Is there a ptas for euclidean kmeans for arbitrary kand dimension d. On clustering financial time series a need for distances.
The np hardness of the wcss criterion in general dimensions when. The clustering algorithms need to be dispersed, since every node within the network possesses only local knowledge and, due to clusterbased routing, communicates out of the cluster via its ch. Pdf nphardness of euclidean sumofsquares clustering. 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. A strongly np hard problem of partitioning a finite set of points of euclidean space into two clusters is considered.
Though understanding that further distance of a cluster increases the sse, i still dont understand why it is needed for kmeans but not for kmedoids. Tsitsiklis y abstract we show that unless pnp, there exists no polynomial time or even pseudopolynomial time algorithm that can decide whether a multivariate polynomial of degree four or higher even. Calculate the between group sum of squares for the data from. In particular, we were not able either to find a polynomialtime algorithm to compute this bound, or to prove that the problem is nphard. This gap in understanding left open the intriguing possibility that the problem might admit a ptas for all k. The aim is to give a selfcontained tutorial on using the sum of squares algorithm for unsupervised learning problems, and in particular in gaussian mixture models. Together with prior work, this implies that the problem is np hard for all p6 2. The formula for the calculation of the between group sum of squares is. Clustering and sum of squares proofs, part 1 windows on. A recent proof of nphardness of euclidean sumofsquares clustering, due to drineas et al.
Pranjal awasthi, moses charikar, ravishankar krishnaswamy, ali kemal sinop submitted on 11 feb 2015. Input sparsity and hardness for robust subspace approximation. Solving the minimum sumofsquares clustering problem by. Keywords clustering sumofsquares complexity 1 introduction clustering is a powerful tool for automated analysis of data. A recent proof of np hardness of euclidean sumofsquares clustering, due to drineas et al. Abstract in the subspace approximation problem, we seek a kdimensional subspace f of rd that minimizes the sum of pth powers of euclidean distances to a given set. Hardness and algorithms euiwoong lee and leonard j. The clustering algorithms need to be dispersed, since every node within the network possesses only local knowledge and, due to clusterbased. Contribute to jeffmintonthesis development by creating an account on github. Thesis research np hardness of euclidean sum ofsquares clustering. Clustering and sum of squares proofs, part 2 windows on. As for the hardness of checking nonnegativity of biquadratic forms, we know of two di erent proofs.
Most quantization methods are essentially based on data clustering algorithms. In this paper we answer this question in the negative and provide the rst hardness of approximation for the euclidean kmeans problem. Hardness of checking maximal magnitude of a sum of a. Nphardness of finding a subset of vertices in a vertexweighted graph. Cambridge core knowledge management, databases and data mining data management for multimedia retrieval by k.
Input sparsity and hardness for robust subspace approximation kenneth l. Np hardness and approximation algorithms for solving euclidean problem of finding a maximum total weight subset of vectors edward gimadi 1 discrete optimization and operations research sobolev institute of mathematics sb ras novosibirsk, russia alexey baburin, nikolai glebov, artem pyatkin discrete optimization and operations research sobolev institute of mathematics sb ras novosibirsk, russia. Thus, there cannot be an algorithm running in time polynomial in kand 1unless p np. Nphardness and approximation algorithms for solving. We convert, within polynomialtime and sequential processing, an np complete problem into a real. Nphardness and approximation algorithms for solving euclidean problem of finding a maximum total weight subset of vectors edward gimadi 1 discrete optimization and operations research sobolev institute of mathematics sb ras novosibirsk, russia alexey baburin, nikolai glebov, artem pyatkin discrete optimization and operations research sobolev institute of mathematics sb ras novosibirsk. We show in this paper that this problem is nphard in general. 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. How to prove the nphardness of this modified set covering problem. 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. The optimal selection of chs is an np hard problem. There are two main strategies for solving 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. The use of multiple measurements in taxonomic problems.
Things are particularly bad in cryptography, where sometimes the hardness assumption made in a paper is a thinly veiled version of the assumption that the construction. This is partly due to the unsuitability of the euclidean distance metric, which is typically used in data mining. Dec 11, 2017 in our next post we will lift this proof to a sum of squares proof for which we will need to define sum of squares proofs. Nphardness of euclidean sumofsquares clustering semantic. Nphardness of deciding convexity of quartic polynomials and. Now, remember that the working memory experiment investigates the relationship between the change in iq and the number of training sessions. On clustering financial time series a need for distances between dependent random variables 1. Constrained distance based clustering for timeseries. 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. Sum of squares error sse cluster evaluation algorithms. Selcuk candan skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Strict monotonicity in the lattice of clusterings ever, from a more general point of view, these results can be used as a base of reference for developing clus. Pdf abstract a recent proof of nphardness of euclidean sumofsquares clustering, due to drineas et al.
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