An O( n2) simplex algorithm for a class of linear programs with tree structure

Abstract

In connection with the optimal design of centralized circuit-free networks linear 0–1 programming problems arise which are related to rooted trees. For each problem the variables correspond to the edges of a given rooted tree T. Each path from a leaf to the root of T, together with edge weights, defines a linear constraint, and a global linear objective is to be maximized. We consider relaxations of such problems where the variables are not restricted to 0 or 1 but are allowed to vary continouosly between these bounds. The values of the optimal solutions of such relaxations may be used in a branch and bound procedure for the original 0–1 problem. While in [10] a primal algorithm for these relaxations is discussed, in this paper, we deal with the dual linear program and present a version of the simplex algorithm for its solution which can be implemented to run in time O( n 2). For balanced trees T this time can be reduced to O( n log n).