One Partition Approximating All ℓ_p-norm Objectives in Correlation Clustering
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This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously O(1)-approximate for all ℓ_p-norms of the disagreement vector. This proves that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. Our algorithm is the first combinatorial approximation algorithm for the ℓ_2-norm objective, and more generally the first combinatorial algorithm for the ℓ_p-norm objective when 2 ≤ p < ∞. It is also faster than all previous algorithms that minimize the ℓ_p-norm of the disagreement vector, with run-time O(n^ω), where O(n^ω) is the time for matrix multiplication on n × n matrices. When the maximum positive degree in the graph is at most Δ, this can be improved to a run-time of O(nΔ^2 log n).
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