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On Computing Centroids According to the p-Norms of Hamming Distance Vectors

Authors Jiehua Chen, Danny Hermelin, Manuel Sorge



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Author Details

Jiehua Chen
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland
Danny Hermelin
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel
Manuel Sorge
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

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Jiehua Chen, Danny Hermelin, and Manuel Sorge. On Computing Centroids According to the p-Norms of Hamming Distance Vectors. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 28:1-28:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.28

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Facility location and clustering
  • Theory of computation → Lower bounds and information complexity
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Strings
  • Clustering
  • Multiwinner Election
  • Hamming Distance

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