In this paper, we consider the colorful @math -center problem, which is a generalization of the well-known @math -center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to find the smallest radius @math , such that with @math balls of radius @math , the desired number of points of each color can be covered. We obtain a constant approximation for this problem in the Euclidean plane. We obtain this result by combining a "pseudo-approximation" algorithm that works in any metric space, and an approximation algorithm that works for a special class of instances in the plane. The latter algorithm uses a novel connection to a certain matching problem in graphs.
Finally, @math -center and @math -median have been generalized in an orthogonal direction, where there are additional constraints on the centers @cite_20 @cite_0 @cite_3 . Again, the issues studied in these generalizations tend to be quite different from the ones we confront here.
@cite_0 @cite_3 @cite_20
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We present new approximation algorithms for the @math -median and @math -means clustering problems. To this end, we obtain small coresets for @math -median and @math -means clustering in general metric spaces and in Euclidean spaces. In @math , these coresets are of size with polynomial dependency on the dimension @math . This leads to @math -approximation algorithms to the optimal @math -median and @math -means clustering in @math , with running time @math , where @math is the number of points. This improves over previous results. We use those coresets to maintain a @math -approximate @math -median and @math -means clustering of a stream of points in @math , using @math space. These are the first streaming algorithms, for those problems, that have space complexity with polynomial dependency on the dimension. Given n demand points in the plane, the p-center problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demand point to its respective nearest supply point. The p-median problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the p-center and the p-median problems relative to both the Euclidean and the rectilinear metrics are NP-hard. In fact, we prove that it is NP-hard even to approximate the p-center problems sufficiently closely. The reductions are from 3-satisfiability. We consider the problem of rotation averaging under the L 1 norm. This problem is related to the classic Fermat-Weber problem for finding the geometric median of a set of points in IRn. We apply the classical Weiszfeld algorithm to this problem, adapting it iteratively in tangent spaces of SO(3) to obtain a provably convergent algorithm for finding the L 1 mean. This results in an extremely simple and rapid averaging algorithm, without the need for line search. The choice of L 1 mean (also called geometric median) is motivated by its greater robustness compared with rotation averaging under the L 2 norm (the usual averaging process). We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global L 1 optimum) and multiple rotation averaging (for which no such proof exists). The algorithm is demonstrated to give markedly improved results, compared with L 2 averaging. We achieve a median rotation error of 0.82 degrees on the 595 images of the Notre Dame image set. The @math -center problem is a classical combinatorial optimization problem which asks to find @math centers such that the maximum distance of any input point in a set @math to its assigned center is minimized. The problem allows for elegant @math -approximations. However, the situation becomes significantly more difficult when constraints are added to the problem. We raise the question whether general methods can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that respects one constraint more. Our constraint of choice is privacy: Here, we are asked to only open a center when at least @math clients will be assigned to it. We show how to combine privacy with several other constraints.