# Approximation Algorithms for Hamming Clustering Problems

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## Abstract

We study Hamming versions of two classical clustering problems. The *Hamming radius p-clustering* problem (HRC) for a set *S* of *k* binary strings, each of length *n*, is to find *p* binary strings of length *n* that minimize the maximum Hamming distance between a string in *S* and the closest of the *p* strings; this minimum value is termed the *p-radius of S* and is denoted by *ϱ*. The related *Hamming diameter p-clustering* problem (HDC) is to split *S* into *p* groups so that the maximum of the Hamming group diameters is minimized; this latter value is called the
*p-diameter of S.*

First, we provide an integer programming formulation of HRC which yields exact solutions in polynomial time whenever *k* and *p* are constant. We also observe that HDC admits straightforward polynomialtime solutions when *k = O*(log *n*) or *p* = 2. Next, by reduction from the corresponding geometric *p*-clustering problems in the plane under the *L* _{1} metric, we show that neither HRC nor HDC can be approximated within any constant factor smaller than two unless P=NP. We also prove that for any *∈* > 0 it is NP-hard to split *S* into at most *pk* ^{1/7-∈} clusters whose Hamming diameter doesn’t exceed the
*p*-diameter. Furthermore, we note that by adapting Gonzalez’ farthest-point clustering algorithm [6], HRC and HDC can be approximated within a factor of two in time *O(pkn)*. Next, we describe a 2^{ O(pϱ/ε) }k^{O(p/ε)} *n* ^{2}-time (1 + ε)-approximation algorithm for HRC. In particular, it runs in polynomial time when *p = O*(1) and *ϱ = O*(log(*k* + *n*)). Finally, we show how to find in *O((n/ε* + *kn* log *n* + *k* ^{2} log
*n*)(2^{ϱ} *k*)^{2/ε}) time a set *L* of *O* log *k)* strings of length *n* such that for each string in *S* there is at least one string in *L* within distance (1 + *ε*) *ϱ*, for any constant 0 < *ε* < 1.

## Keywords

Approximation Algorithm Polynomial Time Planar Graph Vertex Cover Binary String## Preview

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