Solving Multicommodity Flow Problems by an Approximation Scheme

Abstract

We study an approximation scheme to solve minimum cost multicommodity flow problems to a relative accuracy of $\varepsilon\in (0,1]$. The proposed scheme, which we shall call Algorithm ${\cal A}$, is a bisection-based procedure, so it maintains an interval that contains the optimal value. At each iteration, Algorithm ${\cal A}$ defines a block angular sharing problem that is solved to a relative accuracy of $O(\varepsilon)$. The computed solution defines the current approximation to an optimal solution and it is used by Algorithm ${\cal A}$ to throw away half of the current interval. It is shown that when Algorithm ${\cal A}$ no longer shrinks the current interval, the current solution solves the minimum cost multicommodity flow problem to the given accuracy $\varepsilon$. To compute approximate solutions to the sharing problems that Algorithm ${\cal A}$ defines, we propose using Algorithm ${\cal L}$ from [J. Villavicencio and M. D. Grigoriadis, Approximate Lagrangian decomposition with a modified Karmarkar logarithmic potential, in Network Optimization, Lecture Notes in Econom. and Math. Systems 450, Springer, Berlin, 1997, pp. 471--485], but we replace its fixed small step size by one computed using inexact line searches. This modification allows large steps in Algorithm ${\cal L}$. We show that the coordination complexities of Algorithm ${\cal L}$ and Algorithm ${\cal L}$ with inexact line searches are the same if these algorithms are applied to the block angular sharing problems defined by Algorithm ${\cal A}$. This result is also true for block angular sharing problems with convex blocks and with coupling constraints given by nonnegative linear functions.