A rounding technique for the polymatroid membership problem

Abstract

We present an efficient technique for finding a subset which maximizes ω( X) − ϱ( X) over all subsets of a set E, where ω and ϱ are real modular and polymatroid functions respectively, using as a subroutine an algorithm which finds such a set for functions ω , ϱ which are near ω, ϱ respectively. In particular we can choose ω , ϱ to be rational with denominators equal to 12| E| 3 if we can assume, whenever ϱ( X) + ϱ( Y) > ϱ( X ∪ Y) + ϱ( X ∩ Y), that the difference between the two sides is at least one. By applying our technique, we construct an O(| E| 3 r 2) algorithm for the case where ϱ is a matroid rank function.