Many real-life problems can be modeled as optimization or decision problems on graphs. Also, many of those real-life problems are NP-hard. One traditional method to solve these problems is by branch and bound while another method is by graph decompositions. In the 1980's, Robertson and Seymour conceived of two new ways to decompose the graph in order to solve these problems. These ingenious ideas were only by-products of their work proving Wagner's Conjecture. A branch decomposition is one of these ideas. A paper by Arnborg, Lagergren and Seeseshowed that many NP-complete problems can be solved in polynomial time using divide and conquer techniques on input graphs with bounded branchwidth, but a paper by Seymour and Thomas proved that computing an optimal branch decomposition is also NP-complete. Although computing optimal branch decompositions is NP-complete, there is a plethora of theory about branchwidth and branch decompositions. For example, a paper by Seymour and Thomas offered a polynomial time algorithm to compute the branchwidth and optimal branch decomposition for planar graphs. This doctoral research is concentrated on constructing branch decompositions for graphs and using branch decompositions to solve NP-complete problems modeled on graphs. In particular, a heuristic to compute near-optimal branch decompositions is presented and the heuristic is compared to previous heuristics in the subject. Furthermore, a practical implementation of an algorithm given in a paper by Seymour and Thomas for computing optimal branch decompositions of planar graphs is implemented with the addition of heuristics to give the algorithm a "divide and conquer" design. In addition, this work includes a theoretical result relating the branchwidth of planar graphs to their duals, characterizations of branchwidth for Halin and chordal graphs. Also, this work presents an algorithm for minor containment using a branch decomposition and a parallel implementation of the heuristic for general graphs using p-threads.

This thesis examines three specific routing applications. In the first model, the scheduling of home health care providers from their homes, to a set of patients, and then back to their respective homes, is performed both heuristically and optimally for very small instances. The problem is complicated by the presence of multiple depots, time windows, and the scheduling of lunch breaks. It is shown that the problem can be formulated as a mixed integer programming problem and, in very small instances, solved to optimality with a branch-and-cut procedure. To obtain solutions for larger instances, though, a heuristic is shown to have more success. The second application considers the vehicle routing problem with time windows, or VRPTW. The vehicle routing problem involves finding a set of routes starting and ending at a single depot that together visit a set of customers. In the VRPTW, there is an additional constraint requiring that each customer must be visited within a given time window. The best known solution procedures for solving the VRPTW use a set partitioning model with column generation. Within this framework, we present a new approach for generating valid inequalities, specifically k-path cuts, to improve the linear programming relaxation. Computational results are given for the standard library of test instances. In particular, the results include solutions for ten previously unsolved instances. The final application concerns the less-than-truckload, or LTL, trucking industry. An LTL carrier primarily handles shipments that are significantly smaller than the size of a tractor-trailer. Savings are achieved by consolidating shipments into loads at regional terminals and transporting these loads from terminal to terminal. The strategic load plan determines how to route the flow of consolidated loads from origin terminals to destination terminals cost effectively and allowing for certain service standards. To find good solutions to this problem, we apply a dual-ascent procedure to a related uncapacitated network design problem to obtain computational results for three different companies.

There are several ways in which we can characterize classes of graphs. One such way of classifying graphs is by their branchwidth. In working to characterize the class of graphs with branchwidth at most four beta 4 we have found a set of reductions that reduces members of beta 4 to the zero graph. We have also computed several planar members of the obstruction set Ob4 for graphs with branchwidth at most four. This thesis will summarize previous results on branchwidth and reveal the previously mentioned new results.

Due to the structure of the solution set, an exact solution to a linear program cannot be computed by an interior-point algorithm without adding features, such as finite termination procedures, to the algorithm. Finite termination procedures attempt to compute an exact solution in a finite number of steps. The addition of a finite termination procedure enables interior-point algorithms to generate highly accurate solutions for problems in which the ill-conditioning of the Jacobian in the neighborhood of the solution currently precludes such accuracy. The main ingredients of finite termination are activating the procedure, predicting the optimal partition, formulating a simple mathematical model to compute a solution and developing computational techniques to solve the model. Each of these issues are studied in turn in this thesis. First, the current optimal face identification models are extended to bounded variable linear programming problems. Distance to the lower and upper bounds are incorporated into the model to prevent the computed solution from violating the bound constraints. Theory in the literature is extended to the new model. Empirical evidence shows that the proposed model reduces the number of projection attempts needed to find an exact solution. When early termination is the goal, projection from a pure composite Newton step is advocated. However, the cost may exceed the benefits due to the average need of more than one projection attempt to find an exact solution. Variants of Mehrotra's predictor-corrector primal-dual interior-point algorithm provide the foundation for most practical interior-point codes. To take advantage of all available algorithmic information, we investigate the behavior of the Tapia predictor-corrector indicator, which incorporates the corrector step, to identify the optimal partition. Globally, the Tapia predictor-corrector indicator behaves poorly as do all indicators, but locally exhibits fast convergence.

In order to rationally design bacteria to produce large quantities of their precious metabolic by-products, we must first understand the metabolic pathways that lead from a starting material (e.g., glucose) to a metabolic product (e.g., hydrogen). Given a set of reactions, elementary flux modes can be used to mathematically define all metabolic pathways that are stoichiometrically and thermodynamically feasible. The elementary flux modes are the intersections of positive halfspaces, or the vertices of a convex polyhedron. Although linear programing has long been used to examine polyhedra, biologists have yet to use linear programming's full functionality to rationally design bacteria. The simplex method is one relevant linear programming function that pivots from one vertex of a polyhedron to another. The breadth-first search uses a simplex-like pivot to list every possible vertex of a polyhedron. When listing a polyhedron's vertices, it is important not to repeat vertices and thus a robust search algorithm is needed. I have employed balanced trees to speed up the breadth-first search. Specific examples, including the maximization of succinate production, will be investigated and the vertices of the mixed acid fermentation process in E. coli will be enumerated. The open source Matlab code along with a GUI will be provided, as well as links to m-files, which are available on the Internet.

Implementations of the primal-dual approach in solving linear programming problems still face issues in maintaining numerical stability and in attaining high accuracy. The major source of numerical problems occurs during the solving of a highly ill-conditioned linear system within the algorithm. We perform a numerical investigation to better understand the numerical behavior related to the solution accuracy of an implementation of an infeasible primal-dual interior-point (IPDIP) algorithm in LIPSOL, a linear programming solver. From our study, we learned that most test problems can achieve higher than the standard 10-8 accuracy used in practice, and a high condition number of the ill-conditioned coefficient matrix does not solely determine the attainable solution accuracy. Furthermore, we learned that the convergence of the primal residual is usually most affected by numerical errors. Most importantly, early satisfaction of the primal equality constraints is often conducive to eventually achieving high solution accuracy.

The k-restricted 2-factor problem is that of finding a spanning subgraph consisting of disjoint cycles with no cycle of length less than or equal to k. It is a generalization of the well known Hamilton cycle problem and is equivalent to this problem when n2≤k≤n-1 . This paper considers necessary and sufficient conditions, algorithms, and polyhedral conditions for 2-factors in bipartite graphs and restricted 2-factors in bipartite graphs. We introduce a generalization of the necessary and sufficient condition for 4-restricted 2-factors in bipartite graphs to 2k-restricted 2-factors in bipartite graphs of a particular form.

An important approach to solving many discrete optimization problems is to associate the discrete set (over which we wish to optimize) with the 0-1 vectors in a given polyhedron and to derive linear inequalities valid for these 0-1 vectors from a linear inequality system defining the polyhedron. Lovasz and Schrijver (1991) described a family of operators, called the matrix-cut operators, which generate strong valid inequalities, called matrix cuts, for the 0-1 vectors in a polyhedron. This family includes the commutative, semidefinite and division operators; each operator can be applied iteratively to obtain, in n iterations for polyhedra in n-space, the convex hull of 0-1 vectors. We study the complexity of matrix-cut based methods for solving 0-1 integer linear programs. We first prove bounds on the (rank) number of iterations required to obtain the integer hull. We show that the upper bound of n, mentioned above, can be attained in the case of the semidefinite operator, answering a question of Goemans. We also determine the semidefinite rank of the standard linear relaxation of the traveling salesman polytope up to a constant factor. We study the use of the semidefinite operator in solving numerical instances and present results on some combinatorial examples and also on a few instances from the MIPLIB test set. Finally, we examine the lengths of cutting-plane proofs based on matrix cuts. We answer a question of Pudlak on such proofs, and prove an exponential lower bound on the length of cutting-plane proofs based on one class of matrix cuts.

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