当前位置: X-MOL 学术SIAM J. Comput. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Better Guarantees for $k$-Means and Euclidean $k$-Median by Primal-Dual Algorithms
SIAM Journal on Computing ( IF 1.2 ) Pub Date : 2019-10-22 , DOI: 10.1137/18m1171321
Sara Ahmadian , Ashkan Norouzi-Fard , Ola Svensson , Justin Ward

SIAM Journal on Computing, Ahead of Print.
Clustering is a classic topic in optimization with $k$-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best-known algorithm for $k$-means in Euclidean space with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of $9+\epsilon$, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of $k$-means and (2) satisfy the hard constraint that at most $k$ clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard linear programming (LP) relaxation. Our techniques are quite general, and we also show improved guarantees for $k$-median in Euclidean metrics and for a generalization of $k$-means in which the underlying metric is not required to be Euclidean.


中文翻译:

Primal-Dual算法为$ k $-均值和欧几里德$ k $-中位数提供更好的保证

《 SIAM计算杂志》,预印本。
聚类是优化中的经典话题,$ k $ -means是最基本的此类问题之一。在没有任何输入限制的情况下,在欧氏空间中具有可证明保证的$ k $ -means最著名的算法是简单的局部搜索试探法,产生的近似保证为$ 9 + \ epsilon $,该比率是已知的对这种方法要严格。通过提出一种新的原始对偶方法,我们克服了这一障碍,该方法允许我们(1)利用$ k $-均值的几何结构,并且(2)满足严格的约束条件,即最多选择$ k $个聚类而不会降低近似值保证。我们的主要结果是相对于标准线性规划(LP)松弛的6.357逼近算法。我们的技术很一般
更新日期:2019-10-22
down
wechat
bug