Reviews: Multi-Criteria Dimensionality Reduction with Applications to Fairness

Originality: The authors' solve a problem left open in a previous paper and made strictly improve on previous work for approximation algorithms. They do so by giving new insights into structural properties of extreme points of semi-definite programs and more general convex programs. As far as I understand, the algorithms presented in the paper are not substantially new, but the analysis of these algorithms is novel. Quality: The paper seems complete and the proofs appear correct. The paper tackles interesting problems and reaches satisfying conclusions. They essentially close the book on the k 2 case, make significant improvements for k 2 and leave open some questions for structured data.