Approximations of the Convex Hull of Hamiltonian Cycles for Cubic Graphs

Author: Asghar Moeini

Moeini, Asghar, 2016 Approximations of the Convex Hull of Hamiltonian Cycles for Cubic Graphs, Flinders University, School of Computer Science, Engineering and Mathematics

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Combinatorial optimization concerns problems where optimal solutions lie among a, possibly very large, number of discrete alternative solutions. There are many combinatorial optimization problems in literature, with the Travelling Salesman Problem (TSP) arguably the most famous. In this thesis the standard form of TSP will be considered. In particular, suppose there are N cities, and a traveling salesman is going to start from the home city, pass through all the other cities exactly once and return to the home city. Such a travel path is called a tour or a Hamiltonian cycle (HC). The distance between each pair of cities is given, and so for any tour, the tour length is the sum of distances travelled. Hence TSP can be simply thought of as the optimisation problem of identifying the tour of shortest length.

A simple case of TSP is the Hamiltonian Cycle Problem (HCP). In particular, given a graph, we are asked to determine whether it contains at least one HC or not. With respect to this property - Hamiltonicity - graphs possessing HC are called Hamiltonian, and graphs not possessing a HC are called non-Hamiltonian. Hamiltonian cycle problem is known to be an NP-Complete problem. Although HCP can be regarded as a special case of the TSP, some researchers presume that the underlying difficulty of the TSP is, perhaps, hidden in HCP. Hence, HCP can be considered as a “haupt problem" for solving TSP.

This thesis contains three main contributions:

A. An analysis is supplied of Hamiltonian curves in the polytope that is an LP-relaxation of a feasible region of a linear programming problem associated with an embedding of HCP in a Markov decision process.

B. A generic method is developed to discover and generate new equality constraints that can be used to refine feasible regions of LP-relaxations of integer programming problems such as TSP.

C. An embedding of cubic graphs in suitably constructed universal graph is introduced as a method for identifying structural equality constraints.

An indirect method of tackling HCP is by identifying which cubic graphs are non-Hamiltonian. In recent contributions other researchers developed a formulation that we call the “Base Model”. The latter characterizes a polytope containing the ideal polytope that is the convex hull of all Hamiltonian cycles (if any) of the given graph. Thus, if that polytope is empty, the graph is non-Hamiltonian. Unfortunately, while the polytope associated with the Base Model is successful at so identifying all bridge graphs and approximately 18% of non-Hamiltonian non-bridge graphs, it fails on the remaining 82% of the latter. The strength of our contributions B-C, above, is that they refine the Base Model for cubic graphs to achieve 100% success rate with identifying all non-Hamiltonian instances when the number of vertices is 18 or less.

Keywords: Hamiltonian Cycle Problem, Sparse Traveling salesman Problem, Markov Decision Process, Integer Programing

Subject: Mathematics thesis

Thesis type: Doctor of Philosophy
Completed: 2016
School: School of Computer Science, Engineering and Mathematics
Supervisor: Jerzy Filar