A rounding technique for the polymatroid membership problem Original Research Article

We present an efficient technique for finding a subset which maximizes ω(X) − varrho(X) over all subsets of a set E, where ω and varrho are real modular and polymatroid functions respectively, using as a subroutine an algorithm which finds such a set for functions image, image which are near ω, varrho respectively. In particular we can choose image, image to be rational with denominators equal to 12E3 if we can assume, whenever varrho(X) + varrho(Y) > varrho(X union or logical sum Y) + varrho(X ∩ Y), that the difference between the two sides is at least one. By applying our technique, we construct an O(E3r2) algorithm for the case where varrho is a matroid rank function.