Detecting Non-Hamiltonian Graphs by Improved Linear Programs and Graph Reductions

Author: Kieran Clancy

Clancy, Kieran, 2017 Detecting Non-Hamiltonian Graphs by Improved Linear Programs and Graph Reductions, Flinders University, School of Computer Science, Engineering and Mathematics

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This thesis continues a recent line of research seeking to solve the Hamiltonian cycle problem (HCP), with a particular focus on the provision of polynomial-time certificates of non-Hamiltonicity. Two broad approaches are described. First, integer programming formulations of HCP and their relaxations to linear programs (LPs) are considered. If a graph induces infeasibility in such an LP, this implies non-Hamiltonicity. Second, edges or vertices whose removal does not alter a graph’s Hamiltonicity are identified, resulting in a reduced graph which may be easier to solve. To test the effectiveness of our approaches, we consider all non-Hamiltonian non-bridge cubic graphs with up to 20 vertices, the set of which we call NHNB20.

We consider several notable formulations of HCP and compare the effectiveness of their corresponding LPs on NHNB20. In particular, we consider models which we denote by MCF (Claus, 1984), MCF+ (Gouveia and Pires, 2001), SST (Sherali et al., 2006), and the Base Model (Filar et al., 2015.) We find that the Base Model is the most effective of these, solving 477 out of 2099 instances, where the other three models solve only 399 of the instances. We also consider these models in the context of the travelling salesman problem (TSP), a problem closely related to HCP. We introduce an algorithm for producing small but difficult instances of TSP based on HCP instances, and use this method to produce two TSP problem sets, ATSP16A and ATSP16AC. On average, the Base Model performs the strongest on the two problem sets, and we conjecture that the Base Model dominates MCF and MCF+. We consider non-tough graphs as a special case of non-Hamiltonian graphs. We show that MCF, MCF+ and SST are necessarily infeasible for non-tough graphs, and conjecture that the same result holds for the Base Model.

We introduce a framework of graph reductions which maintain Hamiltonicity, and the contexts in which they may be applied. We begin by considering subgraphs that, if present, allow the removal of edges or vertices. Next, the automorphism group of the graph is found to be very useful in identifying redundant edges. We present an algorithm which combines these two approaches iteratively, and show that it successfully reduces many instances of NHNB20 to trivial HCP instances. Of those not reduced to trivial instances, a majority are partially reduced. We show that the Base Model is more effective on the partially reduced graphs than on the original graphs.

We then augment the Base Model with additional linear constraints. Constraints similar to those of SST are included, and then extended to take advantage of the additional variables in the Base Model. We also add constraints which take advantage of forced edges and 3-cuts, and constraints based on spectral properties of Hamiltonian cycles. Finally we consider the sequential application of the augmented Base Model to subgraphs of a given instance in order to test necessary conditions for Hamiltonicity. The combination of all of the above techniques results in certificates of non-Hamiltonicity for 2087 of the 2099 instances in NHNB20.

Keywords: hamiltonian cycle problem, linear programming, travelling salesman problem, graph automorphisms, cubic graphs, non-hamiltonian graphs, graph reductions

Subject: Mathematics thesis

Thesis type: Doctor of Philosophy
Completed: 2017
School: School of Computer Science, Engineering and Mathematics
Supervisor: Michael Haythorpe