A concave link elimination (CLE) procedure and lower bound for concave topology, capacity and flow assignment network design problems

We examine the Concave Topology Capacity and Flow Assignment (TCFA) problem. Only two algorithms in the literature are appropriate for solving TCFA problems with concave link cost functions: Kleinrock and Gerla’s Concave Branch Elimination (CBE) procedure, [26][20], and a greedy link elimination procedure developed by Gersht [21]. However, neither works well in practice. The CBE procedure does not perform well in the context of strongly concave link cost functions. While Gersht’s algorithm performs well, its processing requirements are such that it is applicable for small network design problems only. We present a Concave Link Elimination (CLE) procedure, based on Gersht’s greedy link elimination procedure. Our algorithm is shown to perform at least as well as Gersht’s procedure and to be significantly faster than both the CBE and Gersht procedures. In addition, we formulate a lower bounding problem which we solve using a continuous branch-and-bound procedure to assess the quality of the design procedures.