A technique for obtaining true approximations for k-center with covering constraints

Area of Science:

• Computer Science
• Operations Research
• Algorithmic Game Theory

Background:

• Classical clustering problems are being adapted to include fairness considerations.
• Existing fair clustering models like Colorful k-Center and Fair Robust k-Center introduce additional constraints.
• Prior approximation algorithms for these fair models yielded only pseudo-approximations due to constraint relaxation.

Purpose of the Study:

• To develop a new algorithmic approach for fair clustering problems with covering constraints.
• To achieve true approximation ratios for Colorful k-Center and Fair Robust k-Center.
• To resolve open questions regarding approximation guarantees for these fair clustering variants.

Main Methods:

• Introduced a novel technique to handle covering constraints in fair clustering.
• Developed approximation algorithms for specific fair k-Center variants.
• Analyzed the approximability of Colorful k-Center under varying numbers of colors.

Main Results:

• Achieved a 4-approximation for Colorful k-Center with a constant number of colors, settling an open question.
• Obtained a 4-approximation for Fair Robust k-Center, also resolving an open problem.
• Demonstrated that Colorful k-Center is not finitely approximable with an unbounded number of colors.

Conclusions:

• The new approach provides true approximation guarantees for important fair clustering problems.
• The results establish tight approximation bounds and inapproximability results for fair k-Center variants.
• This work advances the theoretical understanding of fairness in algorithmic clustering.