Approximation algorithms for spherical k-means problem using local search scheme
Approximation algorithms for spherical k-means problem using local search scheme
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DOI：
10.1016/j.tcs.2020.06.029
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发表时间：
2020-07
期刊：
影响因子：
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通讯作者：
D. Zhang;Y. Cheng;Min Li;Yishui Wang;Dachuan Xu
中科院分区：
文献类型：
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作者：
D. Zhang;Y. Cheng;Min Li;Yishui Wang;Dachuan Xu
球面k-均值问题是文本挖掘中研究较多的聚类问题，给出了d维单位球面S d中的一个n点集D和一个整数k≤n，目标是找到一个中心子集S⊂S d且|S|≤k使D中每个点到最近中心的余弦相异度之和最小.我们证明了k均值问题的任何γ逼近算法都可以适用于具有2个γ逼近比的k均值问题。通过利用KMP的经典局部搜索(9+ϵ)近似算法，得出了SKMP的局部搜索(18+ϵ)近似算法。因此，一个有趣的问题出现了，即是否存在直接使用局部搜索方案的SKMP的近似算法。本文提出了一种局部搜索近似算法，并证明了它的性能保证是(2(4+7)+ϵ)。最后，通过数值计算验证了该局部搜索近似算法的有效性。
In the spherical k-means problem (SKMP), which is a well-studied clustering problem in text mining, we are given an n-point set D in d-dimensional unit sphere S d, and an integer k≤ n. The goal is to find a center subset S⊂ S d with| S|≤ k that minimizes the sum of cosine dissimilarity measure for each point in D to the nearest center. We prove that any γ-approximation algorithm for the k-means problem (KMP) can be adapted to the SKMP with 2γ-approximation ratio. It follows that there is a local search (18+ ϵ)-approximation algorithm for the SKMP, by leveraging the classical local search (9+ ϵ)-approximation algorithm for the KMP. Therefore, an interesting problem arises, that is whether there exists an approximation algorithm using local search scheme directly for the SKMP. In this paper, we present a local search approximation algorithm for the SKMP and prove its performance guarantee is (2 (4+ 7)+ ϵ). We also conduct numerical computation to show the efficiency of the local search approximation algorithm by single-swap operation in the end.