Minimizing the sum of the k largest functions in linear time

Authors:
Wlodzimierz Ogryczak;Arie Tamir
Affiliations:
Institute of Control and Computation Engineering, Warsaw University of Technology, 00-665 Warsaw, Poland;School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
Venue:
Information Processing Letters
Year:
2003

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Abstract

Given a collection of n functions defined on Rd, and a polyhedral set Q ⊂ Rd, 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 Tamir [Discrete Appl. Math. 109 (2001) 293-307], Tokuyama [Proc. 33rd Annual ACM Symp. on Theory of Computing, 2001, pp. 75-84] and Kalcsics, Nickel, Puerto and Tamir [Oper. Res. Lett. 31 (1984) 114-127].