Network design arc set with variable upper bounds

In this paper we study the network design arc set with variable upper bounds. This set appears as a common substructure of many network design problems and is a relaxation of several fundamental mixed-integer sets studied earlier independently. In particular, the splittable flow arc set, the unsplittable flow arc set, the single node fixed-charge flow set, and the binary knapsack set are facial restrictions of the network design arc set with variable upper bounds. Here we describe families of strong valid inequalities that cut off all fractional extreme points of the continuous relaxation of the network design arc set with variable upper bounds. Interestingly, some of these inequalities are also new even for the aforementioned restrictions studied earlier. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 17–28 2007