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The relative frequency depicts the proportion of data points that have each value. The frequency tells the number of data points that have each value. Like the histogram, a relative frequency histogram also has the same shape with a horizontal scale (the x-axis), but the vertical scale (the y-axis) is marked with relative frequencies (percentages of the whole) instead of actual frequencies. A relative frequency histogram is a graphical representation of a frequency distribution where the...
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Related Experiment Video

Updated: Sep 30, 2025

ExCYT: A Graphical User Interface for Streamlining Analysis of High-Dimensional Cytometry Data
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Fair colorful k-center clustering.

Xinrui Jia1, Kshiteej Sheth1, Ola Svensson1

  • 1EPFL, Route Cantonale, 1015 Lausanne, Switzerland.

Mathematical Programming
|March 18, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces the colorful k-center problem, aiming for fair service guarantees across different colored groups of points. An efficient algorithm achieves a 3-approximation, nearing the optimal 2-approximation for this generalized clustering problem.

Keywords:
Approximation algorithmsClustering and facility locationFairnessk-center

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Area of Science:

  • Computational geometry
  • Operations research
  • Fairness in algorithms

Background:

  • The k-center problem seeks to minimize the maximum distance from any point to its nearest center.
  • Fairness considerations necessitate similar service guarantees for different groups (colors) of points.
  • Existing algorithms for generalized k-center problems often fail to meet coverage requirements or are limited to specific geometric settings.

Purpose of the Study:

  • To address the colorful k-center problem, balancing clustering objectives with fairness constraints.
  • To develop an efficient approximation algorithm for the colorful k-center problem.
  • To analyze the algorithmic complexity and provide theoretical guarantees for the proposed solution.

Main Methods:

  • Formulating the colorful k-center problem as an optimization task in a metric space.
  • Developing a novel approximation algorithm that handles multiple colored point sets and coverage requirements.
  • Proving strong integrality gap lower bounds for linear programming relaxations of the problem.

Main Results:

  • An efficient approximation algorithm with a guarantee of 3 is presented.
  • The algorithm overcomes the combined challenges of clustering and subset-sum-like problems.
  • Demonstrated that the problem is significantly harder than the classical k-center problem, evidenced by lower bounds.

Conclusions:

  • The developed algorithm provides a near-optimal solution for the colorful k-center problem.
  • The research advances the understanding of fair resource allocation in algorithmic settings.
  • This work offers a foundation for further research into equitable clustering and facility location problems.