On the complexity of locating linear facilities in the plane

Authors:
Nimrod Megiddo;Arie Tamir
Affiliations:
Statistics Department, Tel Aviv University, Tel Aviv, Israel;Statistics Department, Tel Aviv University, Tel Aviv, Israel
Venue:
Operations Research Letters
Year:
1982

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Abstract

We consider the computational complexity of linear facility location problems in the plane, i.e., given n demand points, one wishes to find r lines so as to minimize a certain objective-function reflecting the need of the points to be close to the lines. It is shown that it is NP-hard to find r lines so as to minimize any isotone function of the distances between given points and their respective nearest lines. The proofs establish NP-hardness in the strong sense. The results also apply to the situation where the demand is represented by r lines and the facilities by n single points.