Minimizing the sum of the \(k\) largest functions in linear time.
Given a collection of \(n\) functions defined on \(R^{d},\) and a polyhedral set \(Q\subset R^{d}\), we consider the problem of minimizing the sum of the \(k\) largest functions of the collection over \(Q.\) Specifically we focus on collections of linear functions and several classes of convex, piecewise linear functions which are defined by location models. We present simple linear programming formulations for these optimization models which give rise to linear-time algorithms when the dimension \(d\) is fixed. Our results improve complexity bounds of several problems reported recently by \textit{A. Tamir} [Discrete Appl. Math. 109, 293--307 (2001; Zbl 0986.90018)], \textit{Tokuyama} [Proc. 33rd Annual ACM Symp. on Theory of Computing, 75--84 (2001)] and \textit{J. Kalcsics, S. Nickel, J. Puerto} and \textit{A. Tamir} [Oper. Res. Lett. 30, 149--158 (2002; Zbl 1010.90036)].
scientific article; zbMATH DE number 1857179
scientific article