Approximation Algorithms for Hamming Clustering Problems

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.