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Chvatal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or $\frac{1}{2}$, such cuts are known as $\{0,\frac{1}{2}\}$-cuts. It has been proven by Caprara and Fischetti (1996) that separation of $\{0,\frac{1}{2}\}$-cuts is NP-hard. In this paper, we study ways to separate $\{0,\frac{1}{2}\}$-cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated $\{0,\frac{1}{2}\}$-cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.
Traffic in communication networks fluctuates heavily over time.
Thus, to avoid capacity bottlenecks, operators highly overestimate
the traffic volume during network planning. In this paper we
consider telecommunication network design under traffic uncertainty,
adapting the robust optimization approach of Bertsimas and Sim [2004]. We
present three different mathematical formulations for this problem,
provide valid inequalities, study the computational implications,
and evaluate the realized robustness.
To enhance the performance of the mixed-integer programming solver
we derive robust cutset inequalities generalizing their
deterministic counterparts. Instead of a single cutset inequality
for every network cut, we derive multiple valid
inequalities by exploiting the extra variables available in the
robust formulations. We show that these inequalities define facets
under certain conditions and that they completely describe a projection
of the robust cutset polyhedron if the cutset consists of a single edge.
For realistic networks and live traffic measurements we compare the
formulations and report on the speed up by the valid inequalities.
We study the "price of robustness" and evaluate the
approach by analyzing the real network load. The results show that
the robust optimization approach has the potential to support
network planners better than present methods.
The planning of a communication network is inevitably depending on the quality of both the planning tool and the demand forecast used. In this article, we show exemplarily how the emerging area of Robust Optimization can advance the network planning by a more accurate mathematical description of the demand uncertainty. After a general introduction of the concept and its application to a basic network design problem, we present two applications: multi-layer and mixed-line-rate network design. We conclude with a discussion of extensions of the robustness concept to increase the accuracy of handling uncertainties.