TY - GEN

T1 - Approximating Fair Clustering with Cascaded Norm Objectives

AU - Chlamtác, Eden

AU - Makarychev, Yury

AU - Vakilian, Ali

N1 - Funding Information:
Email: yury@ttic.edu ‡Toyota Technological Institute at Chicago (TTIC). Supported by NSF award CCF-1934843. Email: vakilian@ttic.edu
Funding Information:
∗Department of Computer Science, Ben-Gurion University. The work was done while the author was visiting and supported in part by TTIC. Email: chlamtac@cs.bgu.ac.il †Toyota Technological Institute at Chicago (TTIC). Supported by NSF awards CCF-1718820, CCF-1955173, and CCF-1934843.
Publisher Copyright:
Copyright © 2022 by SIAM.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - We introduce the (p, q)-Fair Clustering problem. In this problem, we are given a set of points P and a collection of different weight functions W. We would like to find a clustering which minimizes the ℓq-norm of the vector over W of the ℓp-norms of the weighted distances of points in P from the centers. This generalizes various clustering problems, including Socially Fair k-Median and k-Means, and is closely connected to other problems such as Densest k-Subgraph and Min k-Union. We utilize convex programming techniques to approximate the (p, q)-Fair Clustering problem for different values of p and q. When p ≥ q, we get an O(k(p-q)/(2pq)), which nearly matches a kΩ((p-q)/(pq)) lower bound based on conjectured hardness of Min k-Union and other problems. When q ≥ p, we get an approximation which is independent of the size of the input for bounded p,q, and also matches the recent O((log n/(log log n))1/p)-approximation for (p, ∞)-Fair Clustering by Makarychev and Vakilian (COLT 2021)

AB - We introduce the (p, q)-Fair Clustering problem. In this problem, we are given a set of points P and a collection of different weight functions W. We would like to find a clustering which minimizes the ℓq-norm of the vector over W of the ℓp-norms of the weighted distances of points in P from the centers. This generalizes various clustering problems, including Socially Fair k-Median and k-Means, and is closely connected to other problems such as Densest k-Subgraph and Min k-Union. We utilize convex programming techniques to approximate the (p, q)-Fair Clustering problem for different values of p and q. When p ≥ q, we get an O(k(p-q)/(2pq)), which nearly matches a kΩ((p-q)/(pq)) lower bound based on conjectured hardness of Min k-Union and other problems. When q ≥ p, we get an approximation which is independent of the size of the input for bounded p,q, and also matches the recent O((log n/(log log n))1/p)-approximation for (p, ∞)-Fair Clustering by Makarychev and Vakilian (COLT 2021)

UR - http://www.scopus.com/inward/record.url?scp=85124639247&partnerID=8YFLogxK

M3 - Conference contribution

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 2664

EP - 2683

BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2022

PB - Association for Computing Machinery

T2 - 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022

Y2 - 9 January 2022 through 12 January 2022

ER -