This paper is motivated by real-life applications of bi-objective optimization. Having many non dominated solutions, one wishes to cluster the Pareto front using Euclidian distances. The p-center problems, both in the discrete and continuous versions, are proven solvable in polynomial time with a common dynamic programming algorithm. Having @math points to partition in @math clusters, the complexity is proven in @math (resp @math ) time and @math memory space for the continuous (resp discrete) @math -center problem. @math -center problems have complexities in @math . To speed-up the algorithm, parallelization issues are discussed. A posteriori, these results allow an application inside multi-objective heuristics to archive partial Pareto Fronts.
We note that an affine 2d PF is a line in @math , clustering is equivalent to 1 dimensional cases. 1-dimension K-means was proven to be solvable in polynomial time with a DP algorithm in @math time and @math space. This complexity was improved for a DP algorithm in @math time and @math space in @cite_5 . This is thus the complexity of K-means in an affine 2d PF. The specific case, already mentioned in the previous section, of the continuous p-center problem with centers on a straight line is more general that the case of an affine 2d PF, with a complexity proven in @math time and @math space by @cite_34 . 2d cases of clustering problems can also be seen as specific cases of three-dimensional (3d) PF, affine 3d PF. Having NP-hard complexities proven for planar cases of clustering, which is the case for k-means, p-median, k-medoids, p-center problems since @cite_44 @cite_1 , it implies that the considered clustering problems are also NP-hard for 3d PF.
@cite_44 @cite_5 @cite_34 @cite_1
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@d can be approximated up to (1 + e)-factor, for an arbitrary small e > 0, using the O(k e2)-rank approximation of A and a constant. This implies, for example, that the optimal k-means clustering of the rows of A is (1 + e)-approximated by an optimal k-means clustering of their projection on the O(k e2) first right singular vectors (principle components) of A. A (j, k)-coreset for projective clustering is a small set of points that yields a (1 + e)-approximation to the sum of squared distances from the n rows of A to any set of k affine subspaces, each of dimension at most j. Our embedding yields (0, k)-coresets of size O(k) for handling k-means queries, (j, 1)-coresets of size O(j) for PCA queries, and (j, k)-coresets of size (log n)O(jk) for any j, k ≥ 1 and constant e e (0, 1 2). Previous coresets usually have a size which is linearly or even exponentially dependent of d, which makes them useless when d n. Using our coresets with the merge-and-reduce approach, we obtain embarrassingly parallel streaming algorithms for problems such as k-means, PCA and projective clustering. These algorithms use update time per point and memory that is polynomial in log n and only linear in d. For cost functions other than squared Euclidean distances we suggest a simple recursive coreset construction that produces coresets of size (MATH) Let P be a set of n points in @math k (P) denote the minimum over all k-flats @math of max peP Dist(p, ). We present an algorithm that computes, for any 0 k (P) from each point of P. The running time of the algorithm is dnO(k e5log(1 e)). The crucial step in obtaining this algorithm is a structural result that says that there is a near-optimal flat that lies in an affine subspace spanned by a small subset of points in P. The size of this "core-set" depends on k and e but is independent of the dimension.This approach also extends to the case where we want to find a k-flat that is close to a prescribed fraction of the entire point set, and to the case where we want to find j flats, each of dimension k, that are close to the point set. No efficient approximation schemes were known for these problems in high-dimensions, when k>1 or j>1. This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called A-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of P-trajectories and A-trajectones. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for P-trajectories and A-trajectories. It shows that projective transformations map P-trajectories into P-trajectories. It presents Karmarkar's interpretation of A-trajectories as steepest descent paths of the objective function (c, x) with respect to the Riemannian geometry ds2 Z?= dx, dx, x2 restricted to the relative interior of the polytope of feasible solutions. P-trajectories of a canonical form linear program are radial projections of A-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar's algorithm. These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that P-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesics (actually distinguished chords) of a geometry isometric to a Hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The A-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesics of a geometry isometric to Euclidean geometry. Received by the editors July 28, 1986 and, in revised form, September 28, 1987 and March 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 90C05; Secondary 52A40, 34A34. Research of the first author partially supported by ONR contract N00014-87-K0214. (D 1989 American Mathematical Society 0002-9947 89 @math .25 per page Multipolynomial resultants provide the most efficient methods known (in terms as asymptoticcomplexity) for solving certain systems of polynomial equations or eliminating variables (, 1988). The resultant of f"1, ..., f"n in K[x"1,...,x"m] will be a polynomial in m-n+1 variables which is zero when the system f"1=0 has a solution in ^m ( the algebraic closure of K). Thus the resultant defines a projection operator from ^m to ^(^m^-^n^+^1^). However, resultants are only exact conditions for homogeneous systems, and in the affine case just mentioned, the resultant may be zero even if the system has no affine solution. This is most serious when the solution set of the system of polynomials has ''excess components'' (components of dimension >m-n), which may not even be affine, since these cause the resultant to vanish identically. In this paper we describe a projection operator which is not identically zero, but which is guaranteed to vanish on all the proper (dimension=m-n) components of the system f"i=0. Thus it fills the role of a general affine projection operator or variable elimination ''black box'' which can be used for arbitrary polynomial systems. The construction is based on a generalisation of the characteristic polynomial of a linear system to polynomial systems. As a corollary, we give a single-exponential time method for finding all the isolated solution points of a system of polynomials, even in the presence of infinitely many solutions, at infinity or elsewhere.