Capacitated Network Design on Outerplanar Graphs

Network design problems model the efficient allocation of resources like routers, optical fibres, roads, canals etc. to effectively construct and operate critical infrastructures. In this thesis, we consider the capacitated network design problem (CapNDP), which finds applications in supply-chain logistics problems and network security. Here, we are given a network and for each edge in the network, several security reinforcement options. In addition, for each pair of nodes in the network, there is a specified level of protection demanded. The objective is to select a minimum-cost set of reinforcements for all the edges so that an adversary with strength less than the protection level of a particular pair of nodes cannot disconnect these nodes. The optimal solution to this problem cannot, in general, be found in reasonable time. One way to tackle such hard problems is to develop approximation algorithms, which are fast algorithms that are guaranteed to find near-optimal solutions; the worst-case ratio between the cost of the solution output by the algorithm and the optimum cost is called the approximation ratio of the algorithm. In this thesis, we investigate CapNDP when the network structure is constrained to belong to a class of graphs called outerplanar graphs. This particular special case was first considered by Carr, Fleischer, Leung and Philips; while they claimed to obtain an approximation ratio arbitrarily close to 1, their algorithm has certain fatal flaws. We build upon some of the ideas they use to approximate CapNDP on general networks to develop a new algorithm for CapNDP on outerplanar graphs. The approximation ratio achieved by our algorithm improves the state-of-the-art by a doubly exponential factor. We also notice that our methods can be applied to a more general class of problems called column-restricted covering integers programs, and be adapted to improve the approximation ratio on more instances of CapNDP if the structure of the network is known. Furthermore, our techniques also yield interesting results for a completely unrelated problem in the area of data structures

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes