Optimizing Polymatroid Functions

TL;DR

This paper introduces a linear programming approach to optimize polymatroid functions under difference constraints, providing structural insights, a polynomial-time algorithm, and bounds on possible extensions.

Contribution

It presents a novel LP-based method for optimizing polymatroid functions, offering new structural results and efficient algorithms for specific constraint types.

Findings

LP technique rederives structural results

Polynomial-time algorithm for simple difference constraints

Lower bounds on extensions of the method

Abstract

We consider a class of optimization problems that involve determining the maximum value that a function in a particular class can attain subject to a collection of difference constraints. We show that a particular linear programming technique, based on duality and projections, can be used to rederive some structural results that were previously established using more ad hoc methods. We then show that this technique can be used to obtain a polynomial-time algorithm for a certain type of simple difference constraints. Finally we give lower bound results that show that certain possible extensions of these results are probably not feasible.