In this work, we study diversity-aware clustering problems where the data

points are associated with multiple attributes resulting in intersecting

groups. A clustering solution need to ensure that a minimum number of cluster

centers are chosen from each group while simultaneously minimizing the

clustering objective, which can be either $k$-median, $k$-means or

$k$-supplier. We present parameterized approximation algorithms with

approximation ratios $1+ frac{2}{e}$, $1+frac{8}{e}$ and $3$ for

diversity-aware $k$-median, diversity-aware $k$-means and diversity-aware

$k$-supplier, respectively. The approximation ratios are tight assuming Gap-ETH

and FPT $neq$ W[2]. For fair $k$-median and fair $k$-means with disjoint

faicility groups, we present parameterized approximation algorithm with

approximation ratios $1+frac{2}{e}$ and $1+frac{8}{e}$, respectively. For

fair $k$-supplier with disjoint facility groups, we present a polynomial-time

approximation algorithm with factor $3$, improving the previous best known

approximation ratio of factor $5$.

This article explores the concept of diversity-aware clustering problems, where data points have multiple attributes and belong to intersecting groups. The goal is to find a clustering solution that selects a minimum number of cluster centers from each group while minimizing the clustering objective. The clustering objective can be measured using different algorithms such as $k$-median, $k$-means, or $k$-supplier. The article presents parameterized approximation algorithms for diversity-aware $k$-median, diversity-aware $k$-means, and diversity-aware $k$-supplier, with tight approximation ratios assuming Gap-ETH and FPT $neq$ W[2]. Additionally, the article introduces parameterized approximation algorithms for fair $k$-median and fair $k$-means with disjoint facility groups, with approximation ratios of +frac{2}{e}$ and +frac{8}{e}$ respectively. Lastly, for fair $k$-supplier with disjoint facility groups, the article presents a polynomial-time approximation algorithm with a factor of $, improving upon the previous best known approximation ratio of factor $.

## Exploring Diversity-Aware Clustering Problems: Innovative Solutions and Ideas

In this article, we delve into the realm of diversity-aware clustering problems, where data points are associated with multiple attributes resulting in intersecting groups. The challenge lies in creating a clustering solution that not only chooses a minimum number of cluster centers from each group but also minimizes the clustering objective, which can be defined as either $k$-median, $k$-means or $k$-supplier.

### Introducing Parameterized Approximation Algorithms

To address this complex problem, we propose the use of parameterized approximation algorithms with impressive approximation ratios. Specifically, we present algorithms with approximation ratios of + frac{2}{e}$, +frac{8}{e}$ and $ for diversity-aware $k$-median, diversity-aware $k$-means, and diversity-aware $k$-supplier respectively.

These approximation ratios are achieved through careful analysis and optimization techniques. We utilize the concepts of Gap-ETH (Gap-Exponential Time Hypothesis) and FPT (Fixed Parameter Tractability) to establish the tightness of these approximation ratios. Our algorithms provide efficient and reliable solutions for diversity-aware clustering problems.

### Fair Clustering with Disjoint Facility Groups

In addition to the aforementioned algorithms, we also propose solutions for fair clustering with disjoint facility groups. Fair clustering aims to ensure equitable representation and distribution of cluster centers among different groups. By extending our parameterized approximation algorithms, we achieve impressive approximation ratios for fair $k$-median and fair $k$-means.

In the case of fair $k$-median with disjoint facility groups, our algorithm guarantees an approximation ratio of +frac{2}{e}$. Similarly, our fair $k$-means algorithm achieves an approximation ratio of +frac{8}{e}$. These ratios highlight our commitment to promoting fairness and balance in diversity-aware clustering problems.

Nevertheless, our focus on fair clustering doesn’t end here. For fair $k$-supplier with disjoint facility groups, we present a groundbreaking polynomial-time approximation algorithm with a factor of 3. This significantly improves upon the previously known best approximation ratio of factor 5.

### Conclusion: A New Era for Diversity-Aware Clustering

In conclusion, our research introduces innovative solutions and ideas to enhance diversity-aware clustering. Through the application of parameterized approximation algorithms, we achieve impressive approximation ratios and address diverse clustering objectives. Furthermore, our focus on fair clustering with disjoint facility groups paves the way for more equitable and balanced clustering solutions.

With our research, we usher in a new era for diversity-aware clustering, where inclusivity and efficiency go hand in hand. As we continue to explore this field, we hope to inspire more groundbreaking advancements and foster a diverse and inclusive data analytics landscape.

The work presented in this content focuses on diversity-aware clustering problems, where data points have multiple attributes and can belong to intersecting groups. The objective of the clustering solution is to choose a minimum number of cluster centers from each group while minimizing the clustering objective, which can be measured using different metrics such as $k$-median, $k$-means, or $k$-supplier.

The authors propose parameterized approximation algorithms with specific approximation ratios for each of the diversity-aware clustering problems. For diversity-aware $k$-median, they present an algorithm with an approximation ratio of + frac{2}{e}$, which means that the solution found by the algorithm is guaranteed to be at most + frac{2}{e}$ times worse than the optimal solution. Similarly, for diversity-aware $k$-means, they provide an algorithm with an approximation ratio of +frac{8}{e}$, and for diversity-aware $k$-supplier, they present an algorithm with an approximation ratio of $.

These approximation ratios are considered tight assuming Gap-ETH and FPT $neq$ W[2], which suggests that it is unlikely to find algorithms with significantly better approximation ratios for these problems. This indicates that the proposed algorithms are quite effective in achieving good solutions.

Additionally, the authors also address fair clustering problems, where fairness is considered by ensuring that cluster centers are chosen fairly among different groups. For fair $k$-median and fair $k$-means with disjoint facility groups, they provide parameterized approximation algorithms with approximation ratios of +frac{2}{e}$ and +frac{8}{e}$, respectively. This implies that these algorithms can find solutions that are close to the optimal solutions while maintaining fairness among different groups.

Furthermore, for fair $k$-supplier with disjoint facility groups, the authors present a polynomial-time approximation algorithm with a factor of $. This is an improvement over the previously known best approximation ratio of factor $, indicating that the proposed algorithm provides a more efficient and effective solution for this particular clustering problem.

Overall, the work presented in this content contributes to the field of diversity-aware and fair clustering by providing parameterized approximation algorithms with tight approximation ratios for various clustering problems. These algorithms can be valuable in practical applications where diversity and fairness are important considerations, such as in social network analysis, recommendation systems, or market segmentation.

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