According to wikipedia, consider $k$-means problem in the plane :

k-means clustering aims to partition the $n$ observations into $k (≤ n)$ sets $S = \{S_1, S_2, \dots, S_k\}$ so as to minimize the within-cluster sum of squares. Formally, the objective is to find: $$\min\sum_{i=1}^k\sum_{x\in S_i} \| x − \mu_i \|^2$$

where $\mu_i$ is the mean of points in $S_i$.

We know that there is constant factor approximation algorithm for $k$-means problem. Now, consider this example that we find $3$-means clustering in the plane:

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But we want modify the above clusters such that we create such clusters that each cluster has a rectangular shape as bellow:

enter image description here

My question is, can we do some modification on any constant factor approximation algorithm for $k$-means such that give us a rectangular partition and constant factor approximation algorithm? Also, is there any paper about this problem? It seems this problem should be well-studied, but I was unable to find any references.

  • $\begingroup$ Related: cstheory.stackexchange.com/questions/51177/… (now deleted) $\endgroup$
    – Neal Young
    Mar 22, 2022 at 12:02
  • $\begingroup$ @NealYoung Is there any paper about this problem? Or something that give us some hints. I search many times but I can't find any related things about this problem. $\endgroup$
    – All
    Mar 22, 2022 at 13:56
  • $\begingroup$ Do you know anyone who can help with this? $\endgroup$
    – All
    Mar 22, 2022 at 15:53
  • $\begingroup$ I think the user @Jut has had similar questions, maybe they can help. $\endgroup$
    – Neal Young
    Mar 22, 2022 at 19:06

1 Answer 1


Assuming P$\ne$NP, there is no such poly-time approximation algorithm.

I assume here that any approximation algorithm must return some feasible $k$-cover, as long as the given input has one. By a $k$-cover of a given set $S$ of points in the plane, I mean a set of $k$ pairwise-disjoint axis-parallel rectangles such that every point in $S$ is in one of the rectangles.

Theorem 1. No such approximation algorithm runs in polynomial time, unless P=NP.

Proof. If there were such an algorithm, it could decide the following decision problem in poly-time: given $(k, S)$, does $S$ have a $k$-cover? (regardless of objective). By Theorem 2 of [1], this decision problem is NP-hard. $~~~\Box$

Remark. The theorem assumes $k$ is part of the input. For any fixed $k$, the problem can be solved in poly-time by exhaustive search, as there are $O(n^4)$ distinct rectangles to consider, so $O(n^{4k})$ possible covers, so there is an exact algorithm that runs in time $n^{4k+O(1)}$.

[1] Ahn, Hee-Kap, et al. "Covering points by disjoint boxes with outliers." Computational Geometry 44.3 (2011): 178-190. https://doi.org/10.1016/j.comgeo.2010.10.002

  • $\begingroup$ It's a great answer. Thank you. $\endgroup$
    – All
    Mar 23, 2022 at 23:02
  • $\begingroup$ Can we hope maybe there is or not Pseudo-polynomial time approximation algorithm with constant factor? $\endgroup$
    – All
    Mar 24, 2022 at 22:37
  • $\begingroup$ I don't think so.. Essentially the problem of approximating your problem is as hard or harder than the $k$-cover problem as defined above, which is NP-hard. Approximating your problem in pseudo-poly time is no easier than solving an NP-hard problem in pseudo-poly time. $\endgroup$
    – Neal Young
    Mar 26, 2022 at 2:39
  • $\begingroup$ Is there any reference about k-cover problem? I search in internet but I not found anything about that. Or how we relate above problem to k-cover? Becuase we try to optimize k-means but k-cover have no spicific objective function according your explination. $\endgroup$
    – ErroR
    May 9, 2022 at 2:23

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