Polynomial approximate discretization of geometric centers in high-dimensional Euclidean space

Vladimir Shenmaier
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Vladimir Shenmaier: Sobolev Institute of Mathematics

Advances in Data Analysis and Classification, 2022, vol. 16, issue 4, No 9, 1039-1067

Abstract: Abstract Many geometric optimization problems can be reduced to choosing points in space (centers) minimizing some objective function which continuously depends on the distances from the chosen centers to given input points. We prove that, for any fixed $$\varepsilon >0$$ ε > 0 , every finite set of points in any-dimensional real space admits a polynomial-size set of candidate centers which can be computed in polynomial time and which contains a $$(1+\varepsilon )$$ ( 1 + ε ) -approximation of each point of space with respect to the Euclidean distances to all the given points. It provides a universal approximation-preserving reduction of any geometric center-based problems whose objective function satisfies a natural continuity-type condition to their discrete versions where the desired centers are selected from a polynomial-size set of candidates. The obtained polynomial upper bound for the size of a universal centers set is supplemented by a theoretical lower bound for this size in the worst case.

Keywords: Geometric optimization; Clustering; Facility location; Euclidean space; High dimensions; Candidate centers; Discretization; 90C27; 68W25; 6207; 90B85 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11634-021-00481-4

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