Simultaneously Approximating All ๐“_p-Norms in Correlation Clustering

Authors Sami Davies , Benjamin Moseley , Heather Newman



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Sami Davies
  • Department of EECS and Simons Institute, University of California at Berkeley, CA, USA
Benjamin Moseley
  • Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA
Heather Newman
  • Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA

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Sami Davies, Benjamin Moseley, and Heather Newman. Simultaneously Approximating All ๐“_p-Norms in Correlation Clustering. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 52:1-52:20, Schloss Dagstuhl โ€“ Leibniz-Zentrum fรผr Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.52

Abstract

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 ๐“_p-norms of the disagreement vector; in other words, a combinatorial O(1)-approximation of the all-norms objective for correlation clustering. This is the first proof that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. In addition, our algorithm is the first combinatorial approximation algorithm for the ๐“โ‚‚-norm objective, and more generally the first combinatorial algorithm for the ๐“_p-norm objective when 1 < p < โˆž. It is also faster than all previous algorithms that minimize the ๐“_p-norm of the disagreement vector, with run-time O(n^ฯ‰), where O(n^ฯ‰) is the time for matrix multiplication on n ร— n matrices. When the maximum positive degree in the graph is at most ฮ”, this can be improved to a run-time of O(nฮ”ยฒ log n).

Subject Classification

ACM Subject Classification
  • Theory of computation โ†’ Approximation algorithms analysis
Keywords
  • Approximation algorithms
  • correlation clustering
  • all-norms
  • lp-norms

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