Note on Combinatorial Optimization with Max-Linear Objective Functions

Chung, S.-J., H.W. Hamacher, F. Maffioli and K.G. Murty, Note on combinatorial optimization with max-linear objective functions, Discrete Applied Mathematics 42 (1993) 139-145. We consider combinatorial optimization problems with a feasible solution set SC(O,I )” specified by a system of linear constraints in O-l variables. Additionally, several cost functions c,,., ,,c~ are given. The max-linear objective function is defined by flx):=max(c’x,...+!‘x: YES} where c?=(c~,.,,,c:) is for q=l,...,p an integer row vector in Iw”. The problem of minimizingflx) over S is called the max-linear combinatorial optimization (MLCO) problem. We will show that MLCO is NP-hard even for the simplest case of S={O,l}” andp=2, and strongly NP-hard for generalp. We discuss the relation to multi-criteria optimization and develop some bounds for MLCO. Correspondence to: Professor S-J. Chung, Department of Industrial Engineering, Seoul National University, Seoul, South Korea. * Supported by NATO Grant RG85/0240. 0166-218X/93/$06.00