License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2019.28
URN: urn:nbn:de:0030-drops-111495
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11149/
Chen, Jiehua ;
Hermelin, Danny ;
Sorge, Manuel
On Computing Centroids According to the p-Norms of Hamming Distance Vectors
Abstract
In this paper we consider the p-Norm Hamming Centroid problem which asks to determine whether some given strings have a centroid with a bound on the p-norm of its Hamming distances to the strings. Specifically, given a set S of strings and a real k, we consider the problem of determining whether there exists a string s^* with (sum_{s in S} d^{p}(s^*,s))^(1/p) <=k, where d(,) denotes the Hamming distance metric. This problem has important applications in data clustering and multi-winner committee elections, and is a generalization of the well-known polynomial-time solvable Consensus String (p=1) problem, as well as the NP-hard Closest String (p=infty) problem.
Our main result shows that the problem is NP-hard for all fixed rational p > 1, closing the gap for all rational values of p between 1 and infty. Under standard complexity assumptions the reduction also implies that the problem has no 2^o(n+m)-time or 2^o(k^(p/(p+1)))-time algorithm, where m denotes the number of input strings and n denotes the length of each string, for any fixed p > 1. The first bound matches a straightforward brute-force algorithm. The second bound is tight in the sense that for each fixed epsilon > 0, we provide a 2^(k^(p/((p+1))+epsilon))-time algorithm. In the last part of the paper, we complement our hardness result by presenting a fixed-parameter algorithm and a factor-2 approximation algorithm for the problem.
BibTeX - Entry
@InProceedings{chen_et_al:LIPIcs:2019:11149,
author = {Jiehua Chen and Danny Hermelin and Manuel Sorge},
title = {{On Computing Centroids According to the p-Norms of Hamming Distance Vectors}},
booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)},
pages = {28:1--28:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-124-5},
ISSN = {1868-8969},
year = {2019},
volume = {144},
editor = {Michael A. Bender and Ola Svensson and Grzegorz Herman},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11149},
URN = {urn:nbn:de:0030-drops-111495},
doi = {10.4230/LIPIcs.ESA.2019.28},
annote = {Keywords: Strings, Clustering, Multiwinner Election, Hamming Distance}
}
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Keywords: |
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Strings, Clustering, Multiwinner Election, Hamming Distance |
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Collection: |
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27th Annual European Symposium on Algorithms (ESA 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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06.09.2019 |
