eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-09-06
28:1
28:16
10.4230/LIPIcs.ESA.2019.28
article
On Computing Centroids According to the p-Norms of Hamming Distance Vectors
Chen, Jiehua
1
Hermelin, Danny
2
Sorge, Manuel
1
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland
Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol144-esa2019/LIPIcs.ESA.2019.28/LIPIcs.ESA.2019.28.pdf
Strings
Clustering
Multiwinner Election
Hamming Distance