arXiv subjects

Michael Haythorpe

Publications and source records attributed to Michael Haythorpe.

At least 19 recordsLinked to original sources

arXiv2025

Simplifications to Guide Monte Carlo Tree Search in Combinatorial Games

We examine a type of modified Monte Carlo Tree Search (MCTS) for strategising in combinatorial games. The modifications are derived by analysing simplified strategies and simplified versions of the underlying game and then using the results to construct an ensemble-type strategy. We present some instances where relative algorithm performance can be predicted from the results in the simplifications, making the approach useful as a heuristic for developing strategies in highly complex games, especially when simulation-type strategies and comparative analyses are largely intractable.

cs.GT↗

arXiv2024

A Systematic Approach to Crossing Numbers of Cartesian Products with Paths

Determining the crossing numbers of Cartesian products of small graphs with arbitrarily large paths has been an ongoing topic of research since the 1970s. Doing so requires the establishment of coincident upper and lower bounds; the former is usually demonstrated by providing a suitable drawing procedure, while the latter often requires substantial theoretical arguments. Many such papers have been published, which typically focus on just one or two small graphs at a time, and use ad hoc arguments specific to those graphs. We propose a general approach which, when successful, establishes the required lower bound. This approach can be applied to the Cartesian product of any graph with arbitrarily large paths, and in each case involves solving a modified version of the crossing number problem on a finite number (typically only two or three) of small graphs. We demonstrate the potency of this approach by applying it to Cartesian products involving all 133 graphs $G$ of orders five or six, and show that it is successful in 128 cases. This includes 60 cases which a recent survey listed as either undetermined, or determined only in journals without adequate peer review.

math.CO↗

arXiv2023

Variants of the Domination Number for Flower Snarks

We consider the flower snarks, a widely studied infinite family of 3--regular graphs. For the Flower snark $J_n$ on $4n$ vertices, it is trivial to show that the domination number of $J_n$ is equal to $n$. However, results are more difficult to determine for variants of domination. The Roman domination, weakly convex domination, and convex domination numbers have been determined for flower snarks in previous works. We add to this literature by determining the independent domination, 2-domination, total domination, connected domination, upper domination, secure Domination and weak Roman domination numbers for flower snarks.

math.CO↗

arXiv2023

Binary Programming Formulations for the Upper Domination Problem

We consider Upper Domination, the problem of finding the minimal dominating set of maximum cardinality. Very few exact algorithms have been described for solving Upper Domination. In particular, no binary programming formulations for Upper Domination have been described in literature, although such formulations have proved quite successful for other kinds of domination problems. We introduce two such binary programming formulations, and show that both can be improved with the addition of extra constraints which reduce the number of feasible solutions. We compare the performance of the formulations on various kinds of graphs, and demonstrate that (a) the additional constraints improve the performance of both formulations, and (b) the first formulation outperforms the second in most cases, although the second performs better for very sparse graphs. Also included is a short proof that the upper domination number of any generalized Petersen graph P(n,k) is equal to n.

math.CO↗

arXiv2023

A Cross Entropy Approach to the Domination Problem and its Variants

The domination problem and several of its variants (total domination, 2-domination and secure domination) are considered. These problems have various real-world applications, but are NP-hard to solve to provable optimality, making fast heuristics for these problems desirable. There is a wealth of highly-developed heuristics and approximation algorithms for the domination problem, however such heuristics are much less common for variants of the domination problem. We redress this by proposing an implementation of the cross entropy method that can be applied to any sensible variant of domination. We present results from experiments which demonstrate that this approach can produce good results in an efficient manner even for larger graphs, and that it works roughly as well for any of the domination variants considered.

math.OC↗

arXiv2021

A survey of graphs with known or bounded crossing numbers

We present, to the best of the authors' knowledge, all known results for the (planar) crossing numbers of specific graphs and graph families. The results are separated into various categories; specifically, results for general graph families, results for graphs arising from various graph products, and results for recursive graph constructions.

math.CO↗

arXiv2021

The Secure Domination Number of Cartesian Products of Small Graphs with Paths and Cycles

The secure domination numbers of the Cartesian products of two small graphs with paths or cycles is determined, as well as for Mobius ladder graphs. Prior to this work, in all cases where the secure domination number has been determined, the proof has either been trivial, or has been derived from lower bounds established by considering different forms of domination. However, the latter mode of proof is not applicable for most graphs, including those considered here. Hence, this work represents the first attempt to determine secure domination numbers via the properties of secure domination itself, and it is expected that these methods may be used to determine further results in the future.

math.CO↗

arXiv2021

Finding a Hamiltonian cycle by finding the global minimizer of a linearly constrained problem

It has been shown that a global minimizer of a smooth determinant of a matrix function corresponds to the largest cycle of a graph. When it exists, this is a Hamiltonian cycle. Finding global minimizers even of a smooth function is a challenge. The difficulty is often exacerbated by the existence of many global minimizers. One may think this would help, but in the case of Hamiltonian cycles the ratio of the number of global minimizers to the number of local minimizers is typically astronomically small. There are various equivalent forms of the problem and here we report on two. Although the focus is on finding Hamiltonian cycles, and this has an interest in and of itself, this is just a proxy for a class of problems that have discrete variables. The solution of relaxations of these problems is typically at a degenerate vertex, and in the neighborhood of the solution the Hessian is indefinite. The form of the Hamiltonian cycle problem we address has the virtue of being an ideal test problem for algorithms designed for discrete nonlinear problems in general. It is easy to generate problems of varying size and varying character, and they have the advantage of being able to determine if a global solution has been found. A feature of many discrete problems is that there are many solutions. For example, in the frequency assignment problem any permutation of a solution is also a solution. A consequence is that a common characteristic of the relaxed problems is that they have large numbers of global minimizers and even larger numbers of both local minimizers, and saddle points whose reduced Hessian has only a single negative eigenvalue. Efficient algorithms that seek to find global minimizers for this type of problem are described. Results using BONMIN, a solver for nonlinear problems with continuous and discrete variables, are also included.

math.OC↗

arXiv2021

Constructing Families of Cospectral Regular Graphs

A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid for special cases of a property introduced by Schwenk. For the case of cubic (3-regular) graphs, computational results are given which show that the construction generates a large proportion of the cubic graphs, which are cospectral with another cubic graph.

math.CO↗

arXiv2021

On the Crossing Numbers of Cartesian Products of Small Graphs with Paths, Cycles and Stars

There has been significant research dedicated towards computing the crossing numbers of families of graphs resulting from the Cartesian products of small graphs with arbitrarily large paths, cycles and stars. For graphs with four or fewer vertices, these have all been computed, but there are still various gaps for graphs with five or more vertices. We contribute to this field by determining the crossing numbers for fifteen such families.

math.CO↗

arXiv2020

The maximum crossing number of $C_3 \times C_3$

We determine that the maximum crossing number of $C_3 \times C_3$ is 78, which closes the previously best known range of between 68 and 80. The proof uses several techniques which may be useful in determining the maximum crossing number of other graphs.

math.CO↗

arXiv2019

An improved binary programming formulation for the secure domination problem

The secure domination problem, a variation of the domination problem with some important real-world applications, is considered. Very few algorithmic attempts to solve this problem have been presented in literature, and the most successful to date is a binary programming formulation which is solved using CPLEX. A new binary programming formulation is proposed here which requires fewer constraints and fewer binary variables than the existing formulation. It is implemented in CPLEX, and tested on certain families of graphs that have previously been considered in the context of secure domination. It is shown that the runtime required for the new formulation to solve the instances is significantly less than that of the existing formulation. An extension of our formulation that solves the related, but further constrained, secure connected domination problem is also given; to the best of the authors' knowledge, this is the first such formulation in literature.

math.CO↗

arXiv2019

There are no Cubic Graphs on 26 Vertices with Crossing Number 10 or 11

We show that no cubic graphs of order 26 have crossing number larger than 9, which proves a conjecture of Ed Pegg Jr and Geoffrey Exoo that the smallest cubic graphs with crossing number 11 have 28 vertices. This result is achieved by first eliminating all girth 3 graphs from consideration, and then using the recently developed QuickCross heuristic to find good embeddings of each remaining graph. In the cases where the embedding found has 10 or more crossings, the heuristic is re-run with a different settings of parameters until an embedding with fewer than 10 crossings is found. We provide a minimal example of a cubic graph on 28 vertices with crossing number 10, and also exhibit for the first time a cubic graph on 30 vertices with crossing number 12, which we conjecture is minimal.

math.CO↗

arXiv2019

Deterministic "Snakes and Ladders" Heuristic for the Hamiltonian Cycle Problem

We present a polynomial complexity, deterministic, heuristic for solving the Hamiltonian Cycle Problem (HCP) in an undirected graph of order $n$. Although finding a Hamiltonian cycle is not theoretically guaranteed, we have observed that the heuristic is successful even in cases where such cycles are extremely rare, and it also performs very well on all HCP instances of large graphs listed on the TSPLIB web page. The heuristic owes its name to a visualisation of its iterations. All vertices of the graph are placed on a given circle in some order. The graph's edges are classified as either snakes or ladders, with snakes forming arcs of the circle and ladders forming its chords. The heuristic strives to place exactly $n$ snakes on the circle, thereby forming a Hamiltonian cycle. The Snakes and Ladders Heuristic (SLH) uses transformations inspired by $k-$opt algorithms such as the, now classical, Lin-Kernighan heuristic to reorder the vertices on the circle in order to transform some ladders into snakes and vice versa. The use of a suitable stopping criterion ensures the heuristic terminates in polynomial time if no improvement is made in $n^3$ major iterations.

math.CO↗

arXiv2019

A New Heuristic for Detecting Non-Hamiltonicity in Cubic Graphs

We analyse a polyhedron which contains the convex hull of all Hamiltonian cycles of a given undirected connected cubic graph. Our constructed polyhedron is defined by polynomially-many linear constraints in polynomially-many continuous (relaxed) variables. Clearly, the emptiness of the constructed polyhedron implies that the graph is non-Hamiltonian. However, whenever a constructed polyhedron is non-empty, the result is inconclusive. Hence, the following natural question arises: if we assume that a non-empty polyhedron implies Hamiltonicity, how frequently is this diagnosis incorrect? We prove that, in the case of bridge graphs, the constructed polyhedron is always empty. We also demonstrate that some non-bridge non-Hamiltonian cubic graphs induce empty polyhedra as well. We compare our approach to the famous Dantzig-Fulkerson-Johnson relaxation of a TSP, and give empirical evidence which suggests that the latter is infeasible if and only if our constructed polyhedron is also empty. By considering special edge cut sets which are present in most cubic graphs, we describe a heuristic approach, built on our constructed polyhedron, for which incorrect diagnoses of non-Hamiltonian graphs as Hamiltonian appear to be very rare. In particular, for cubic graphs containing up to 18 vertices, only four out of 45,982 undirected connected cubic graphs were so misdiagnosed. By constrast, we demonstrate that an equivalent heuristic, when built on the Dantzig-Fulkerson-Johnson relaxation of a TSP, is mostly unsuccessful in identifying additional non-Hamiltonian graphs. These empirical results suggest that polynomial algorithms based on our constructed polyhedron may be able to correctly identify Hamiltonicity of a cubic graph in all but rare cases.

math.CO↗

arXiv2019

Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

It is well known that 3--regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3--regular graphs without reducing the girth, thereby proving that such graphs with arbitrarily large girth also exist. The resulting graphs can be 1--, 2-- or 3--edge-connected depending on the construction chosen. From the constructions arise (naive) upper bounds on the size of the smallest non-Hamiltonian 3--regular graphs with particular girth. Several examples are given of the smallest such graphs for various choices of girth and connectedness.

math.CO↗

arXiv2019

On the Determinant and its Derivatives of the Rank-one Corrected Generator of a Markov Chain on a Graph

We present an algorithm to find the determinant and its first and second derivatives of a rank-one corrected generator matrix of a doubly stochastic Markov chain. The motivation arises from the fact that the global minimiser of this determinant solves the Hamiltonian cycle problem. It is essential for algorithms that find global minimisers to evaluate both first and second derivatives at every iteration. Potentially the computation of these derivatives could require an overwhelming amount of work since for the Hessian $N^2$ cofactors are required. We show how the doubly stochastic structure and the properties of the objective may be exploited to calculate all cofactors from a single LU decomposition.

math.OC↗

arXiv2019

Linearly-growing Reductions of Karp's 21 NP-complete Problems

We address the question of whether it may be worthwhile to convert certain, now classical, NP-complete problems to one of a smaller number of kernel NP-complete problems. In particular, we show that Karp's classical set of 21 NP-complete problems contains a kernel subset of six problems with the property that each problem in the larger set can be converted to one of these six problems with only linear growth in problem size. This finding has potential applications in optimisation theory because the kernel subset includes 0-1 integer programming, job sequencing and undirected Hamiltonian cycle problems.

math.CO↗