An Oracle Strongly Polynomial Algorithm for Bottleneck Expansion Problems

• Published in: Optimization methods & software Vol. 17; no. 1; pp. 61 - 75
• Main Authors: Zhang, Jianzhong, Liu, Zhenhong
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01.02.2002
• Subjects: Bottleneck Capacity; Capacity Expansion; Polynomially Solvable
• ISSN: 1055-6788; 1029-4937
• DOI: 10.1080/10556780290027819

Let E = { e 1 , e 2 , , e n } be a finite set and $\cal F $ be a family of subsets of E . For each element e i in E , c i is a given capacity and w i is the cost of increasing capacity c i by one unit. The problem is how to expand the capacities of elements in E so that the capacity of $\cal F $ is as large as possible, subject to a given budget restriction. This problem was introduced in [1] where an algorithm was proposed which is polynomial under some conditions. However, that method is not strongly polynomial. In this paper this problem is solved by solving a combinatorial equation. It is shown that if the problem $ \min _{F\in {\cal F}}w(F) $ is solvable in strongly polynomial time, then the bottleneck expansion problem is also solvable in strongly polynomial time. This result is stronger than what the method in [1] gives. In addition, some interesting variations of this problem are also discussed.