### 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. 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. We provide an integer programming formulation of HRC which yields exact solutions in polynomial time whenever k is constant. We also observe that HDC admits straightforward polynomial-time solutions when k = O(logn) and p = O(1), or when 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 ε > 0 it is NP-hard to split S into at most pk^{1/7-ε} clusters whose Hamming diameter does not exceed the p-diameter, and that solving HDC exactly is an NP-complete problem already for p = 3. Furthermore, we note that by adapting Gonzalez' farthest-point clustering algorithm [T. Gonzalez, Theoret. Comput. Sci. 38 (1985) 293 306], 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) (2ρk)^{2/ε}) time a set L of O(p 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.

Original language | English (US) |
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Pages (from-to) | 289-301 |

Number of pages | 13 |

Journal | Journal of Discrete Algorithms |

Volume | 2 |

Issue number | 2 SPEC. ISS. |

DOIs | |

State | Published - Jun 2004 |

Externally published | Yes |

Event | Combinatiorial Pattern Matching - Montreal, Canada Duration: Jun 21 2000 → Jun 23 2000 |

### Keywords

- Approximation algorithms
- Hamming distance
- Integer programming
- NP-hardness
- P-clustering problem

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Discrete Algorithms*,

*2*(2 SPEC. ISS.), 289-301. https://doi.org/10.1016/S1570-8667(03)00079-0