## No file available

To read the file of this research,

you can request a copy directly from the authors.

Preprint

Preprints and early-stage research may not have been peer reviewed yet.

This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian Completion Problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in the context of various specific kinds of undirected graphs (e.g. trees, unicyclic graphs and series-parallel graphs). The proposed algorithm, however, concentrates on solving HCP for general undirected graphs. It can be considered to belong to the category of matheuristics, because it integrates an exact linear time solution for trees into a local search algorithm for general graphs. This integration makes use of the close relation between HCP and the minimum path partition problem, which makes the algorithm equally useful for solving the latter problem. Furthermore, a benchmark set of problem instances is constructed for demonstrating the quality of the proposed algorithm. A comparison with state-of-the-art solvers indicates that the proposed algorithm is able to achieve high-quality results.

To read the file of this research,

you can request a copy directly from the authors.

ResearchGate has not been able to resolve any citations for this publication.

Matheuristics are heuristic algorithms made by the interoperation of metaheuristics and mathematic programming (MP) techniques. An essential feature is the exploitation in some part of the algorithms of features derived from the mathematical model of the problems of interest, thus the definition “model-based metaheuristics” appearing in the title of some events of the conference series dedicated to matheuristics [1]. The topic has attracted the interest of a community of researchers, and this led to the publication of dedicated volumes and journal special issues, [13], [14], besides to dedicated tracks and sessions on wider scope conferences.
The increasing maturity of the area permits to outline some trends and possibilities offered by matheuristic approaches. A word of caution is needed before delving into the subject, because obviously the use of MP for solving optimization problems, albeit in a heuristic way, is much older and much more widespread than matheuristics. However, this is not the case for metaheuristics, and also the very idea of designing MP methods specifically for heuristic solution has innovative traits, when opposed to exact methods which turn into heuristics when enough computational resources are not available.

We consider a machine with a single real variable x that describes its state. Jobs J1, ', JN are to be sequenced on the machine. Each job requires a starting state A, and leaves a final state Bi. This means that Ji can be started only when x = Ai and, at the completion of the job, x = Bi. There is a cost, which may represent time or money, etc., for changing the machine state x so that the next job may start. The problem is to find the minimal cost sequence for the N jobs. This problem is a special case of the traveling salesman problem. We give a solution requiring only 0N2 simple steps. A solution is also provided for the bottleneck form of this traveling salesman problem under special cost assumptions. This solution permits a characterization of those directed graphs of a special class which possess Hamiltonian circuits.

This paper analyzes the asymptotic worst-case running time of a number,of variants of the well-known method,of path compression,for maintaining a collection of disjoint sets under union. We show that two one-pass methods,proposed by van Leeuwen and van der Weide are asymptotically optimal, whereas several other methods, including one proposed by Rein and advocated by Dijkstra, are slower than the best methods. Categories and Subject Descriptors: E. 1 [Data Structures]: Trees; F2.2 [Analysis of Algorithms and

Large networks are becoming a widely used abstraction for studying complex systems in a broad set of disciplines, ranging from social-network analysis to molecular biology and neuroscience. Despite an increasing need to analyze and manipulate large networks, only a limited number of tools are available for this task.
Here, we describe the Stanford Network Analysis Platform (SNAP), a general-purpose, high-performance system that provides easy-to-use, high-level operations for analysis and manipulation of large networks. We present SNAP functionality, describe its implementational details, and give performance benchmarks. SNAP has been developed for single big-memory machines, and it balances the trade-off between maximum performance, compact in-memory graph representation, and the ability to handle dynamic graphs in which nodes and edges are being added or removed over time. SNAP can process massive networks with hundreds of millions of nodes and billions of edges. SNAP offers over 140 different graph algorithms that can efficiently manipulate large graphs, calculate structural properties, generate regular and random graphs, and handle attributes and metadata on nodes and edges. Besides being able to handle large graphs, an additional strength of SNAP is that networks and their attributes are fully dynamic; they can be modified during the computation at low cost. SNAP is provided as an open-source library in C++ as well as a module in Python.
We also describe the Stanford Large Network Dataset, a set of social and information real-world networks and datasets, which we make publicly available. The collection is a complementary resource to our SNAP software and is widely used for development and benchmarking of graph analytics algorithms.

Arthur Cayley (1821-1895) was a key figure in the creation of modern algebra. He studied mathematics at Cambridge and published three papers while still an undergraduate. He then qualified as a lawyer and published about 250 mathematical papers during his fourteen years at the Bar. In 1863 he took a significant salary cut to become the first Sadleirian Professor of Pure Mathematics at Cambridge, where he continued to publish at a phenomenal rate on nearly every aspect of the subject, his most important work being in matrices, geometry and abstract groups. In 1883 he became president of the British Association for the Advancement of Science. Publication of his Collected Papers - 967 papers in 13 volumes plus an index volume - began in 1889 and was completed after his death by his successor in the Sadleirian Chair. This volume contains 76 papers published between 1856 and 1862, plus one from 1891.

Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mech-anisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

This book presents the latest findings on one of the most intensely investigated subjects in computational mathematics--the traveling salesman problem. It sounds simple enough: given a set of cities and the cost of travel between each pair of them, the problem challenges you to find the cheapest route by which to visit all the cities and return home to where you began. Though seemingly modest, this exercise has inspired studies by mathematicians, chemists, and physicists. Teachers use it in the classroom. It has practical applications in genetics, telecommunications, and neuroscience. The authors of this book are the same pioneers who for nearly two decades have led the investigation into the traveling salesman problem. They have derived solutions to almost eighty-six thousand cities, yet a general solution to the problem has yet to be discovered. Here they describe the method and computer code they used to solve a broad range of large-scale problems, and along the way they demonstrate the interplay of applied mathematics with increasingly powerful computing platforms. They also give the fascinating history of the problem--how it developed, and why it continues to intrigue us.

The Hamiltonian completion problem for an arbitrary graph. G. is the determination of the smallest number of new edges which must be added to G to make the resulting graph Hamilltonian. For general graphs, the associated decision problem is NP-complete although polynomial time algorithms do exist for special cases. In this paper, we review the known results concerning Hamiltonion completion and develop a suite of exact and approximate solution algorithms. Guidelines are given as to which of the various algorithms should be used in given circumstances

A minimum Hamiltonian completion of a graph G is a minimum-size set of edges that, when added to G, guarantee a Hamiltonian path. Finding a Hamiltonian completion has applications to frequency assignment as well as distributed computing. If the new edges are deleted from the Hamiltonian path, one is left with a minimum path cover, a minimum-size set of vertex-disjoint paths that cover the vertices of G. For arbitrary graphs, constructing a minimum Hamiltonian completion or path cover is clearly NP-hard, but there exists a linear-time algorithm for trees. In this paper we first give a description and proof of correctness for this linear-time algorithm that is simpler and more intuitive than those given previously. We show that the algorithm extends also to unicyclic graphs. We then give a new method for finding an optimal path cover or Hamiltonian completion for a tree that uses a reduction to a maximum flow problem. In addition, we show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles, that is, a 2-factor.

We define the Hamiltonian completion number of a graph G, denoted hc(G), to be the minimum number of lines that need to be added to G in order to make it Hamiltonian. The Hamiltonian
completion problem asks for hc(G) and a specific Hamiltonian cycle containing hc(G) new lines. We derive an efficient algorithm
for finding hc(T) for any tree T, and show that if S is the set of spanning trees of an arbitrary connected graph G, then
hc(G) = minTi eS hc(Ti ).hc(G) = \mathop {min}\limits_{T_i \varepsilon S} hc(T_i ).
.
A number of other general results are presented including an efficient heuristic procedure which can be used on arbitrary
graphs.

Given a graph G=(V,E), the Hamiltonian completion number ofG, HCN(G), is the minimum number of edges to be added to G to make it Hamiltonian. This problem is known to be
NP\mathcal{NP}
-hard even when G is a line graph. In this paper, local search algorithms for finding HCN(G) when G is a line graph are proposed. The adopted approach is mainly based on finding a set of edge-disjoint trails that dominates
all the edges of the root graph ofG. Extensive computational experiments conducted on a wide set of instances allow to point out the behavior of the proposed
algorithms with respect to both the quality of the solutions and the computation time.

This report describes an implementation of the Lin-Kernighan heuristic, one of the most successful methods for generating optimal or nearoptimal solutions for the symmetric traveling salesman problem. Computational tests show that the implementation is highly effective. It has found optimal solutions for all solved problem instances we have been able to obtain, including a 7397-city problem (the largest nontrivial problem instance solved to optimality today). Furthermore, the algorithm has improved the best known solutions for a series of large-scale problems with unknown optima, among these an 85900-city problem.

The probability that a random graph with n vertices and cn log n edges contains a Hamiltonian circuit tends to 1 as n → ∞ (if c is sufficiently large).

Givena (directed or un directed) graph G, finding the smallest number of additional edges which make the graph Hamiltonianis called the HamiltonianCompletionProblem (HCP). We consider this problem in the context of sparse random graphs G(n, c/n) on n nodes, where each edge is selected independently with probability c/n. We give a complete asymptotic answer to this problem when c< 1, by constructing a new linear time algorithm for solving HCP on trees and by using generating function method. We solve the problem both in the cases of undirected and directed graphs. © 2005 Elsevier B.V. All rights reserved.

Given a graph G=(V,E),HCN(L(G)) is the minimum number of edges to be added to its line graph L(G) to make L(G) Hamiltonian. This problem is known to be NP-hard for general graphs, whereas a O(|V|) algorithm exists when G is a tree. In this paper a linear algorithm for finding HCN(L(G)) when G is a cactus is proposed.

A series-parallel graph can be constructed from a certain graph by recurslvely applying "series" and "parallel" connections The class of such graphs, which Is a well-known model of series-parallel electrical networks, is a subclass of planar graphs It is shown in a umfied manner that there exist hnear- ume algorithms for many combinatorial problems ff an input graph is restricted to the class of series-parallel graphs. These include 0) the decision problem with respect to a property characterized by a finite number of forbidden graphs, (u) the mlmmum edge (vertex) deletion problem with respect to the same property as above, and (Ul) the generalized matching problem Consequently, the following problems, among others, prove to be hnear-tlme computable for the class of series-parallel graphs. (I) the minimum vertex cover problem, (2) the maximum outerplanar (reduced) subgraph problem, (3) the minimum feedback vertex set problem, (4) the maximum (induced) hne-subgraph problem, (5) the maximum matching problem, and (6) the maximum disjoint triangle problem.

The Hamiltonian completion problem for an arbitrary graph G consists of determining the minimum number of new lines which can be added to G in order to produce a Hamiltonian cycle in G. A solution to this problem would be useful in situations where it is necessary to periodically traverse a network or data structure in such a way as to visit all nodes and minimize the length of the traversal. Linear algorithms are presented for solving the Hamiltonian completion problem for several classes of graphs, in particular for trees and unicyclic graphs. Several more general results are also given.

Given a graph G=(V,E), HCN(L(G)) is the minimum number of edges to be added to its line graph L(G) to make L(G) Hamiltonian. This problem is known to be NP-hard for general graphs, whereas an O(|V|5) algorithm exists when G is a tree. In this paper a linear algorithm for finding HCN(L(G)) when G is a tree is proposed.

This text is an in-depth account of graph theory. It reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains descriptive passages designed to convey the flavour of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory such as colouring, matching, extremal theory, and algebraic graph theory, the book presents an account of newer topics, including: Szemer'edi's Regularity Lemma and its use; Shelah's extension of the Hales-Jewett Theorem; the precise nature of the phase transition in a random graph process; the connection between electrical networks and random walks on graphs; and the Tutte polynomial and its cousins in knot theory.

- D A Bader
- H Meyerhenke
- P Sanders
- D Wagner

D. A. Bader, H. Meyerhenke, P. Sanders, and D. Wagner, editors. Graph Partitioning and Graph Clustering, 10th
DIMACS Implementation Challenge Workshop, Georgia Institute of Technology, Atlanta, GA, USA, February 13-14, 2012. Proceedings, volume 588 of Contemporary Mathematics. American Mathematical Society, 2013.

Using POPMUSIC for Candidate Set Generation in the Lin-Kernighan-Helsgaun TSP Solver

- K Helsgaun

K. Helsgaun. Using POPMUSIC for Candidate Set Generation in the Lin-Kernighan-Helsgaun TSP Solver.
Roskilde Universitet, 7 2018.