A General Framework for Designing Approximation Schemes for Combinatorial Optimization Problems with Many Objectives Combined into One

In this paper, we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function. Examples of such problems include those in which the objective function is a product or ratio of two linear functions, parallel machine scheduling problems with the makespan objective, robust versions of weighted multi-objective optimization problems, and assortment optimization problems with logit choice models. The main idea behind our approximation schemes is the construction of an approximate Pareto-optimal front of the functions which constitute the given objective. Using this idea, we give the first fully polynomial time approximation schemes for the max-min resource allocation problem with a fixed number of agents and for combinatorial optimization problems in which the objective function is the sum of a fixed number of ratios of linear functions, or the product of a fixed number of linear functions. ∗Operations Research Scientist, Amazon.com, 331 Boren Avenue N, Seattle, WA 98107. This work was done when the author was a graduate student in the Operations Research Center at Massachusetts Institute of Technology. Email: [email protected]. ∗∗Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, E62-569, Cambridge MA 02139, USA. Email: [email protected].