Inspired by prior work by Tian and by Cao and Xu, this paper presents an efficient computer-aided framework to characterize the fundamental limits of coded caching systems under the constraint of linear coding. The proposed framework considers non-Shannon-type inequalities which are valid for representable polymatroids (and hence for linear codes), and leverages symmetric structure and problem-specific constraints of coded caching to reduce the complexity of the linear program. The derived converse bounds are tighter compared to previous known analytic methods, and prove the optimality of some achievable memory-load tradeoff points under the constraint of linear coding placement and delivery. These results seem to indicate that small, structured demand subsets combined with minimal common information constructions may be sufficient to characterize optimal tradeoffs under linear coding.

Abstract. Biobjective mixed integer linear programs (BOMILP) are optimization problems where two linear objectives are optimized over a polyhedron while restricting some of the variables to be integer. Since many of the techniques for solving BOMILP (or approximating its solution set) are iterative processes which utilize data discovered during early iterations to aid in the discovery of improved data during later iterations, it is highly desirable to efficiently store the nondominated subset of a given set of data. This problem has not received considerable attention in the context of BOMILP; only naive methods have been implemented. We seek to bridge this gap by presenting a new data structure in the form of a modified binary tree that stores, updates, searches and returns nondominated solutions. This structure takes points and line segments in R2 as input and stores the nondominated subset of this input. We note that when used alongside an exact solution procedure, such as branch-and-bound (BB), at termination the data stored by this structure is precisely the set of Pareto optimal solutions. We perform two experiments. The first is designed to compare the utility of our structure for storing nondominated data to that of a dynamic list which updates via pairwise comparison. In the second we use our data structure alongside the biobjective BB techniques available in the literature and solve specific instances of BOMILP. The results of our first experiment suggest that the data structure performs reasonably well in handling input of up to 107 points or segments and does so much more efficiently than a dynamic list. The results of the second experiment show that when our structure is utilized alongside BB fathoming is enhanced and running times improve slightly. 1.

Abstract We investigate a network design problem under traffic uncertainty that arises when provisioning Virtual Private Networks (VPNs): given a set of terminals that must communicate with one another, and a set of possible traffic matrices, sufficient capacity has to be reserved on the links of the large underlying public network to support all possible traffic matrices while minimizing the total reservation cost. The problem admits several versions depending on the desired topology of the reserved links, and the nature of the traffic data uncertainty. We present compact linear mixed‐integer programming formulations for the problem with the classical hose traffic model and for a less conservative robust variant relying on the traffic statistics that are often available. These flow‐based formulations allow us to solve optimally medium‐to‐large instances with commercial MIP solvers. We also propose a combined branch‐and‐price and cutting‐plane algorithm to tackle larger instances. Computational results obtained for several classes of instances are reported and discussed. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 49(1), 100–115 2007

Abstract Although there is an extensive literature dealing with network design, little attention has been devoted to networks with complicated node costs. Although node costs, depending linearly on the total passing flow, can be easily embedded into the more usual framework of networks with link costs, when the node costs are, for instance, a stepwise function of the facilities installed into the nodes, this is no longer possible. This feature seems to be crucial in modern telecommunications networks, but has also applications in other fields, where a limited set of technologies is available with discrete values of capacities and costs. In our specific application, we propose a mathematical programming model that explicitly accounts for node costs that are stepwise with nonlinear increments. Two families of valid inequalities are then introduced, one of which is an extension of those presented in a previous work by Stoer and Dahl for multifacility network models. As the separation problem for these inequalities is difficult, we develop a heuristic separation procedure. We devise a branch‐and‐cut method and test it on a set of real‐world instances found in the network design literature. This new method proves to be efficient when compared to a commercial general purpose MIP algorithm. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 49(1), 90–99 2007