The inverse 1-median problem on a cycle

Authors:
Rainer E. Burkard;Carmen Pleschiutschnig;Jianzhong Zhang
Affiliations:
Institute of Optimization and Discrete Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria;Institute of Optimization and Discrete Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria;Department of Mathematics, City University of Hong Kong, Hong Kong
Venue:
Discrete Optimization
Year:
2008

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Abstract

Let the graph G=(V,E) be a cycle with n+1 vertices, non-negative vertex weights and positive edge lengths. The inverse 1-median problem on a cycle consists in changing the vertex weights at minimum cost so that a prespecified vertex becomes the 1-median. All cost coefficients for increasing or decreasing the weights are assumed to be 1. We show that this problem can be formulated as a linear program with bounded variables and a special structure of the constraint matrix: the columns of the linear program can be partitioned into two classes in which they are monotonically decreasing. This allows one to solve the problem in O(n^2) time.