Separating a superclass of comb inequalities in planar graphs

Many classes of valid and facet-inducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2-matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x*, finds one or more inequalities in the class which are violated by x*, or proves that none exist. Such algorithms are at the core of the highly successful branch-and-cut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2-matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We define a generalization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges with x*_e > 0 is planar. The separation algorithm runs in O(n^3) time, where n is the number of vertices in the graph

Tree-based decompositions of graphs on surfaces and applications to the traveling salesman problem

The tree-width and branch-width of a graph are two well-studied examples of parameters that measure how well a given graph can be decomposed into a tree structure. In this thesis we give several results and applications concerning these concepts, in particular if the graph is embedded on a surface. In the first part of this thesis we develop a geometric description of tangles in graphs embedded on a fixed surface (tangles are the obstructions for low branch-width), generalizing a result of Robertson and Seymour. We use this result to establish a relationship between the branch-width of an embedded graph and the carving-width of an associated graph, generalizing a result for the plane of Seymour and Thomas. We also discuss how these results relate to the polynomial-time algorithm to determine the branch-width of planar graphs of Seymour and Thomas, and explain why their method does not generalize to surfaces other than the sphere. We also prove a result concerning the class C_2k of minor-minimal graphs of branch-width 2k in the plane, for an integer k at least 2. We show that applying a certain construction to a class of graphs in the projective plane yields a subclass of C_2k, but also show that not all members of C_2k arise in this way if k is at least 3. The last part of the thesis is concerned with applications of graphs of bounded tree-width to the Traveling Salesman Problem (TSP). We first show how one can solve the separation problem for comb inequalities (with an arbitrary number of teeth) in linear time if the tree-width is bounded. In the second part, we modify an algorithm of Letchford et al. using tree-decompositions to obtain a practical method for separating a different class of TSP inequalities, called simple DP constraints, and study their effectiveness for solving TSP instances.Ph.D.Committee Chair: Thomas, Robin; Committee Co-Chair: Cook, William J.; Committee Member: Dvorak, Zdenek; Committee Member: Parker, Robert G.; Committee Member: Yu, Xingxin

Primal Cutting Plane Methods for the Traveling Salesman Problem

Most serious attempts at solving the traveling salesman problem (TSP) are based on the dual fractional cutting plane approach, which moves from one lower bound to the next. This thesis describes methods for implementing a TSP solver based on a primal cutting plane approach, which moves from tour to tour with non-degenerate primal simplex pivots and so-called primal cutting planes. Primal cutting plane solution of the TSP has received scant attention in the literature; this thesis seeks to redress this gap, and its findings are threefold. Firstly, we develop some theory around the computation of non-degenerate primal simplex pivots, relevant to general primal cutting plane computation. This theory guides highly efficient implementation choices, a sticking point in prior studies. Secondly, we engage in a practical study of several existing primal separation algorithms for finding TSP cuts. These algorithms are all conceptually simpler, easier to implement, or asymptotically faster than their standard counterparts. Finally, this thesis may constitute the first computational study of the work of Fleischer, Letchford, and Lodi on polynomial-time separation of simple domino parity inequalities. We discuss exact and heuristic enhancements, including a shrinking-style heuristic which makes the algorithm more suitable for application on large-scale instances. The theoretical developments of this thesis are integrated into a branch-cut-price TSP solver which is used to obtain computational results on a variety of test instances

GAPS : a hybridised framework applied to vehicle routing problems

In this thesis we consider two combinatorial optimisation problems; the Capacitated Vehicle Routing Problem (CVRP) and the Capacitated Arc Routing Problem (CARP). In the CVRP, the objective is to find a set of routes for a homogenous fleet of vehicles, which must service a set of customers from a central depot. In contrast, the CARP requires a set of routes for a fleet of vehicles to service a set of customers at the street level of an intercity network. After a comprehensive discussion of the existing exact and heuristic algorithmic techniques presented in the literature for these problems, computational experiments to provide a benchmark comparison of a subset of algorithmic implementations for these methods are presented for both the CVRP and CARP, run against a series of dataset instances from the literature. All dataset instances are re-catalogued using a standard format to overcome the difficulties of the different naming schemes and duplication of instances that exist between different sources. We then present a framework, which we shall call Genetic Algorithm with Perturbation Scheme (GAPS), to solve a number of combinatorial optimisation problems. The idea is to use a genetic algorithm as a container framework in conjunction with a perturbation or weight coding scheme. These schemes make alterations to the underlying input data within a problem instance, after which the changed data is fed into a standard problem specific heuristic and the solution obtained decoded to give a true solution cost using the original unaltered instance data. We first present GAPS in a generic context, using the Travelling Salesman Problem (TSP) as an example and then provide details of the specific application of GAPS to both the CVRP and CARP. Computational experiments on a large set of problem instances from the literature are presented and comparisons with the results achieved by the current state of the art algorithmic approaches for both problems are given, highlighting the robustness and effectiveness of the GAPS framework.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

GAPS: a hybridised framework applied to vehicle routing problems

In this thesis we consider two combinatorial optimisation problems; the Capacitated Vehicle Routing Problem (CVRP) and the Capacitated Arc Routing Problem (CARP). In the CVRP, the objective is to find a set of routes for a homogenous fleet of vehicles, which must service a set of customers from a central depot. In contrast, the CARP requires a set of routes for a fleet of vehicles to service a set of customers at the street level of an intercity network. After a comprehensive discussion of the existing exact and heuristic algorithmic techniques presented in the literature for these problems, computational experiments to provide a benchmark comparison of a subset of algorithmic implementations for these methods are presented for both the CVRP and CARP, run against a series of dataset instances from the literature. All dataset instances are re-catalogued using a standard format to overcome the difficulties of the different naming schemes and duplication of instances that exist between different sources. We then present a framework, which we shall call Genetic Algorithm with Perturbation Scheme (GAPS), to solve a number of combinatorial optimisation problems. The idea is to use a genetic algorithm as a container framework in conjunction with a perturbation or weight coding scheme. These schemes make alterations to the underlying input data within a problem instance, after which the changed data is fed into a standard problem specific heuristic and the solution obtained decoded to give a true solution cost using the original unaltered instance data. We first present GAPS in a generic context, using the Travelling Salesman Problem (TSP) as an example and then provide details of the specific application of GAPS to both the CVRP and CARP. Computational experiments on a large set of problem instances from the literature are presented and comparisons with the results achieved by the current state of the art algorithmic approaches for both problems are given, highlighting the robustness and effectiveness of the GAPS framewor