One and Two Facility Network Design Revisited

Abstract

The one facility one commodity network design problem (OFOC) with nonnegative flow costs considers the problem of sending d units of flow from a source to a destination where arc capacity is purchased in batches of C units. The two facility problem (TFOC) is similar, but capacity can be purchased either in batches of C units or one unit. Flow costs are zero. These problems are known to be NP-hard. We describe an exact O(n ³3ⁿ) algorithm for these problems based on the repeated use of a bipartite matching algorithm. We also present a better lower bound of Ω(n ² k ^*) for an earlier Ω(n ^2k) algorithm described in the literature where k=⌊d/C⌋ and k ^*=min k,⌊(n−2)/2⌋. The matching algorithm is faster than this one for k≥⌊(n−2)/2⌋. Finally, we provide another reformulation of the problem that is quasi integral. This property could be useful in designing a modified version of the simplex method to solve the problem using a sequence of pivots with integer extreme solutions, referred to as the integral simplex method in the literature.