Simultaneous Optimization via Approximate Majorization for Concave Profits or Convex Costs

AbstractFor multicriteria problems and problems with a poorly characterized objective, it is often desirable to approximate simultaneously the optimum solution for a large class of objective functions. We consider two such classes: (1) Maximizing all symmetric concave functions. (2) Minimizing all symmetric convex functions. The first class corresponds to maximizing profit for a resource allocation problem (such as allocation of bandwidths in a computer network). The concavity requirement corresponds to the law of diminishing returns in economics. The second class corresponds to minimizing cost or congestion in a load balancing problem, where the congestion/cost is some convex function of the loads. Informally, a simultaneous α-approximation for either class is a feasible solution that is within a factor α of the optimum for all functions in that class. Clearly, the structure of the feasible set has a significant impact on the best possible α and the computational complexity of finding a solution that achieves (or nearly achieves) this α. We develop a framework and a set of techniques to perform simultaneous optimization for a wide variety of problems. We first relate simultaneous α-approximation for both classes to α-approximate majorization. Then we prove that α-approximately majorized solutions exist for logarithmic values of α for the concave profits case. For both classes, we present a polynomial-time algorithm to find the best α if the set of constraints is a polynomial-sized linear program and discuss several non-trivial applications. These applications include finding a (log n)-majorized solution for multicommodity flow, and finding approximately best α for various forms of load balancing problems. Our techniques can also be applied to produce approximately fair versions of the facility location and bi-criteria network design problems. In addition, we demonstrate interesting connections between distributional load balancing (where the sizes of jobs are drawn from known probability distributions but the actual size is not known at the time of placement) and approximate majorization.