Fair clustering bound tightened: α-core emptiness proven below 2.1508
New proof improves lower bound by 7.5%, narrowing a 7-year gap.
Deep Dive
Cookson, Deltl, and Oh improved lower bounds for proportionally fair clustering by presenting an instance where the α-core is empty for every α < 2.1508, improving the previous lower bound of 2. They used a Mixed Integer Linear Program (MILP) to search for optimal lower-bound instances. For clustering instances with m centers where m is 3, 4, 5, or 6, they determined exact thresholds α_m^* such that an α_m^*-core always exists, while every smaller α has an empty-core instance.
Key Points
- Improved lower bound for α-core emptiness from 2 to 2.1508, closing 7-year gap
- Used MILP to search for optimal counterexamples; exact thresholds for m=3–6 centers
- Techniques connect Hare and Droop cores, reducing problem to structured instances
Why It Matters
Tighter fairness guarantees in clustering lead to more equitable AI systems for resource allocation and representation.