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Title: On Approximability of 𝓁₂² Min-Sum Clustering
The 𝓁₂² min-sum k-clustering problem is to partition an input set into clusters C_1,…,C_k to minimize ∑_{i=1}^k ∑_{p,q ∈ C_i} ‖p-q‖₂². Although 𝓁₂² min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate 𝓁₂² min-sum k-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the 𝓁₂² min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a fast PTAS for 𝓁₂² min-sum k-clustering. Specifically, our algorithm runs in time O(n^{1+o(1)}d⋅ 2^{(k/ε)^O(1)}), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error α ∈ [0,1/2). We give a polynomial-time algorithm that outputs a (1+γα)/(1-α)²-approximation to 𝓁₂² min-sum k-clustering, for a fixed constant γ > 0.  more » « less
Award ID(s):
2443697
PAR ID:
10646612
Author(s) / Creator(s):
; ; ; ;
Editor(s):
Aichholzer, Oswin; Wang, Haitao
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
332
ISSN:
1868-8969
Page Range / eLocation ID:
62:1-62:18
Subject(s) / Keyword(s):
Clustering hardness of approximation polynomial-time approximation schemes learning-augmented algorithms Theory of computation → Computational geometry Theory of computation → Facility location and clustering
Format(s):
Medium: X Size: 18 pages; 1010695 bytes Other: application/pdf
Size(s):
18 pages 1010695 bytes
Sponsoring Org:
National Science Foundation
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