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I have a clustering problem with $\{0,1\}$ distances between a set of nodes, which can be stated as follows:

Given: Finite set $\mathbb{X}$, a distance $d(x, y) \in \{0,1\}$ for each pair $(x, y) \in \mathbb{X}$, and positive integers $S$ and $H$.

QUESTION: Is there a partition of $\mathbb{X}$ into disjoint sets $\mathbb{X}_1, \mathbb{X}_2, \dots, \mathbb{X}_S$ to minimize $\sum_{s=1}^S \sum_{x,y \in \mathbb{X}_s} d(x,y)$ where $|\mathbb{X}_s| \leq H$?

Is there a specific name for such a clustering problem so that I can find the complexity proof of it?

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This is a variant of the minimum $k$-cut problem. The node set is $\mathbb{X}$, the edge weights are $1-d(x,y)$, and $k=S$.

Also related to the wedding planner problem, where $\mathbb{X}$ is the set of guests, $S$ is the number of tables, and $H$ is the maximum number of guests per table. Here, $d(x,y)$ measures whether $x$ and $y$ dislike each other. See https://blogs.sas.com/content/operations/2014/11/10/do-you-have-an-uncle-louie-optimal-wedding-seat-assignments/

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  • $\begingroup$ Thank you, professor, for you answer. $\endgroup$
    – OR Junior
    Commented Jan 13, 2022 at 14:21
  • $\begingroup$ Since it is a SAS employee linking to a SAS blog post....Disclosure: This excellent answer provided by the Senior R&D Manager, SAS Institute Inc. $\endgroup$ Commented Jan 13, 2022 at 14:27

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