Tractability through approximation : a study of two discrete optimization problems

Abstract

(cont.) algorithm, at one extreme, and complete enumeration, at the other extreme. We derive worst-case approximation guarantees on the solution produced by such an algorithm for matroids. We then define a continuous relaxation of the original problem and show that some of the derived bounds apply with respect to the relaxed problem. We also report on a new bound for independence systems. These bounds extend, and in some cases strengthen, previously known results for standard best-in greedy.

This dissertation consists of two parts. In the first part, we address a class of weakly-coupled multi-commodity network design problems characterized by restrictions on path flows and 'soft' demand requirements. In the second part, we address the abstract problem of maximizing non-decreasing submodular functions over independence systems, which arises in a variety of applications such as combinatorial auctions and facility location. Our objective is to develop approximate solution procedures suitable for large-scale instances that provide a continuum of trade-offs between accuracy and tractability. In Part I, we review the application of Dantzig-Wolfe decomposition to mixed-integer programs. We then define a class of multi-commodity network design problems that are weakly-coupled in the flow variables. We show that this problem is NP-complete, and proceed to develop an approximation/reformulation solution approach based on Dantzig-Wolfe decomposition. We apply the ideas developed to the specific problem of airline fleet assignment with the goal of creating models that incorporate more realistic revenue functions. This yields a new formulation of the problem with a provably stronger linear programming relaxation, and we provide some empirical evidence that it performs better than other models proposed in the literature. In Part II, we investigate the performance of a family of greedy-type algorithms to the problem of maximizing submodular functions over independence systems. Building on pioneering work by Conforti, Cornu6jols, Fisher, Jenkyns, Nemhauser, Wolsey and others, we analyze a greedy algorithm that incrementally augments the current solution by adding subsets of arbitrary variable cardinality. This generalizes the standard best-in greedy