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Parameterized Approximation Schemes for Fair-Range Clustering
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Gives the first FPT(k, ℓ) (1+ε)-approximation schemes for fair-range k-median and k-means in Euclidean spaces, combining coreset/JL data reduction with annular net-based candidate selection.
Problem
Fair-range k-median and k-means in Euclidean spaces, where opened facilities must satisfy per-label lower and upper bounds; exact FPT solutions are W[2]-hard and general-metric FPT approximations are limited to constant or 1+2/e ratios.
Approach
Extends the annulus-based framework of Cohen-Addad et al. by partitioning each annulus into cells via gamma-nets over Euclidean facilities, selecting net points as candidates, and reducing dimension to O(log k + log log n) with a coreset plus terminal JL embedding so the running time is FPT in k and ℓ.
Key results
- Randomized (1+ε)-approximation for fair-range k-median in Euclidean spaces in O(d log d) + 2^{(kε^{-1})^{O(1)}+kℓ} n^{O(1)} time.
- Analogous (1+ε)-approximation for fair-range k-means (ratio 1+83√ϵ with ϵ=(ε/83)^2) with the same running time form.
- First parameterized approximation schemes for these Euclidean problems, improving on the O(1)-approximations of Thejaswi et al. (KDD 2022).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply Euclidean net-based facility selection and data reduction | Fair-range k-median · Euclidean fair-range k-median | approximation ratio: O(1) -> (1 + ε) | Helped |
| Apply Euclidean net-based facility selection and data reduction | Fair-range k-means · Euclidean fair-range k-means | approximation ratio: O(1) -> (1 + ε) | Helped |
Limitations
- Running time is exponential in k and ℓ, so it is only FPT and not practical for large k or ℓ.
- Algorithm is randomized and succeeds with constant probability (about 1-1/e), requiring repetition for higher confidence.
- No experiments are reported.
Experimental setup
Metrics: approximation ratio running time
Baselines: Thejaswi et al. 2022 FPT approximation algorithms
Builds on: Cohen-Addad et al. 2019 annulus framework Thejaswi et al. 2022 FPT approximation for fair-range clustering Chen 2009 coreset construction Johnson-Lindenstrauss transform
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