A rounding technique for the polymatroid membership problem

• Published in: Linear algebra and its applications Vol. 221; pp. 41 - 57
• Main Author: Narayanan, H.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.1995
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/0024-3795(93)00222-L

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.