On computing centroids according to the p-norms of hamming distance vectors

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 (Ʃ_s∈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 = ∞) 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 ∞. Under standard complexity assumptions the p reduction also implies that the problem has no 2°^(n+m)-time or 2°^{(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 ε > 0, we provide a 2^{k(p/p+1) +ε}-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.