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 binary strings have a centroid with a bound on the $p$-norm of its Hamming distances to the strings. Specifically, given a set of strings $S$ and a real $k$, we consider the problem of determining whether there exists a string $s^{\ast }$ with $({\sum }_{s\in S}{d}^p(s^{\ast },s){)}^{1/p}\le k$, where $d(,)$ denotes the Hamming distance metric. This problem has important applications in data clustering, and is a generalization of the well-known polynomial-time solvable \textsc{Consensus String} $(p=1)$ problem, as well as the NP-hard \textsc{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 $2^{o(k^{\frac{p}{(p+1)}})}$-time algorithm, where $m$ denotes the number of input strings and $n$ denotes the length of each string, for any fixed $p>1$. Both running time lower bounds are tight. In particular, we provide a $2^{k^{\frac{p}{(p+1)}+\epsilon }}$-time algorithm for each fixed $\epsilon >0$. 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.

To appear at ESA 2019