Browsing by Author "Cook, William J."
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Item A computational study of vehicle routing applications(1999) Rich, Jennifer Lynn; Cook, William J.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.Item Branch decompositions and their applications(2000) Hicks, Illya VaShun; Cook, William J.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.Item Linear-time algorithms for graphs with bounded branchwidth(2003) Christian, William Anderson, Jr; Dean, Nathaniel; Cook, William J.We present an algorithmic framework (including a single data structure) that is extended into linear-time algorithms to solve several NP-complete graph problems (i.e., INDEPENDENT SET, M AXIMUM CUT, GRAPH COLORING, HAMILTONIAN CYCLE, and DISJOINT PATHS). The linearity is achieved assuming the provision of a branch decomposition of the instance graph. We then modify the framework to create a multithreaded framework that uses the existing problem-specific extensions without any revision. Computational results for the serial and parallel algorithms are provided. In addition, we present a graphical package called JPAD that can display a graph and branch decomposition, show their relationship to each other, and be extended to rim and display the progress and results of algorithms on graphs or on branch decompositions.Item On the matrix cuts of Lovasz and Schrijver and their use in integer programming(2001) Dash, Sanjeeb; Cook, William J.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.