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MINIMUM DIAMETER COVERING PROBLEMS Esther M. Arkin y and Refael Hassin z
 

Summary: MINIMUM DIAMETER COVERING PROBLEMS
Esther M. Arkin y and Refael Hassin z
May 20, 1997
Abstract A set V and a collection of (possibly non­disjoint) subsets are given. Also given is a real matrix
describing distances between elements of V . A cover is a subset of V containing at least one representative
from each subset. The multiple­choice minimum diameter problem is to select a cover of minimum diameter.
The diameter is defined as the maximum distance of any pair of elements in the cover. The multiple­choice
dispersion problem, which is closely related, asks us to maximize the minimum distance between any pair of
elements in the cover. The problems are NP­hard. We present polynomial time algorithms for approximating
special cases and generalizations of these basic problems, and we prove in other cases that no such algorithms
exist (assuming P 6= NP ).
1 Introduction
This paper deals with a class of multiple­choice problems. We assume that a set V of elements is given,
together with subsets S 1 ; :::; Sm . A cover is a subset of V that contains at least one representative from each
S j ; j = 1; :::; m.
Our interest is in problems in which the objective is to select a cover whose elements are close to each
other with respect to a given distance function. Our measure of proximity is the diameter of the set. The
diameter is defined as the maximum distance of any pair of elements in the cover. Thus, the multiple­choice
cover problem is to compute a cover of minimum diameter.
An illustrative way to describe our basic problem is as follows: There are m firms (or government offices,

  

Source: Arkin, Estie - Department of Applied Mathematics and Statistics, SUNY at Stony Brook

 

Collections: Mathematics