A Local Search Approximation Algorithm for the k-means Problem with Penalties

Abstract

In this paper, we study the k-means problem with (nonuniform) penalties (k-MPWP) which is a natural generalization of the classic k-means problem. In the k-MPWP, we are given an n-client set $\mathcal {D} \subset \mathbb {R}^d$, a penalty cost $p_j>0$ for each $j \in \mathcal {D}$, and an integer $k \le n$. The goal is to open a center subset $F \subset \mathbb {R}^d$ with $|F| \le k$ and to choose a client subset $P \subseteq \mathcal {D}$ as the penalized client set such that the total cost (including the sum of squares of distance for each client in $\mathcal {D} \setminus P$ to the nearest open center and the sum of penalty cost for each client in P) is minimized. We offer a local search $( 81+ \varepsilon )$-approximation algorithm for the k-MPWP by using single-swap operation. We further improve the above approximation ratio to $( 25+ \varepsilon )$ by using multi-swap operation.