Abstract
We study the classical metric kmedian clustering problem over a set of input rankings (i.e., permutations), which has myriad applications, from socialchoice theory to web search and databases. A folklore algorithm provides a 2approximate solution in polynomial time for all k = O(1), and works irrespective of the underlying distance measure, so long it is a metric; however, going below the 2factor is a notorious challenge. We consider the Ulam distance, a variant of the wellknown editdistance metric, where strings are restricted to be permutations. For this metric, Chakraborty, Das, and Krauthgamer [SODA, 2021] provided a (2δ)approximation algorithm for k = 1, where δ≈ 2^{40}.
Our primary contribution is a new algorithmic framework for clustering a set of permutations. Our first result is a 1.999approximation algorithm for the metric kmedian problem under the Ulam metric, that runs in time (k log (nd))^{O(k)} nd³ for an input consisting of n permutations over [d]. In fact, our framework is powerful enough to extend this result to the streaming model (where the n input permutations arrive one by one) using only polylogarithmic (in n) space. Additionally, we show that similar results can be obtained even in the presence of outliers, which is presumably a more difficult problem.
BibTeX  Entry
@InProceedings{chakraborty_et_al:LIPIcs.ITCS.2023.31,
author = {Chakraborty, Diptarka and Das, Debarati and Krauthgamer, Robert},
title = {{Clustering Permutations: New Techniques with Streaming Applications}},
booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
pages = {31:131:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772631},
ISSN = {18688969},
year = {2023},
volume = {251},
editor = {Tauman Kalai, Yael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17534},
URN = {urn:nbn:de:0030drops175340},
doi = {10.4230/LIPIcs.ITCS.2023.31},
annote = {Keywords: Clustering, Approximation Algorithms, Ulam Distance, Rank Aggregation, Streaming}
}
Keywords: 

Clustering, Approximation Algorithms, Ulam Distance, Rank Aggregation, Streaming 
Collection: 

14th Innovations in Theoretical Computer Science Conference (ITCS 2023) 
Issue Date: 

2023 
Date of publication: 

01.02.2023 