Hamiltonian Completion Research Articles

Evolving test instances of the Hamiltonian completion problem

Predicting and comparing algorithm performance on graph instances is challenging for multiple reasons. First, there is not always a standard set of instances to benchmark performance. Second, using existing graph generators results in a restricted spectrum of difficulty and the resulting graphs are not always diverse enough to draw sound conclusions. That is why recent work proposes a new methodology to generate a diverse set of instances by using evolutionary algorithms. We can then analyze the resulting graphs and get key insights into which attributes are most related to algorithm performance. We can also fill observed gaps in the instance space in order to generate graphs with previously unseen combinations of features. We apply this methodology to the instance space of the Hamiltonian completion problem using two different solvers, namely the Concorde TSP Solver and a multi-start local search algorithm.

A multi-start local search algorithm for the Hamiltonian completion problem on undirected graphs

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.

Evolutionary operators for the Hamiltonian completion problem

This paper deals with evolutionary algorithms for solving the Hamiltonian completion problem. More precisely, the paper is concerned with a collection of crossover and mutation operators, which mostly originate from the traveling salesman problem, but have further on been modified or customized for Hamiltonian completion. The considered crossovers and mutations are tested on a set of randomly generated problem instances. The obtained experimental results clearly show that the behavior and relative ranking of the operators within the Hamiltonian completion environment are different than within the traveling salesman environment. Moreover, it is shown that our modified or custom-designed operator variants accomplish much better results for Hamiltonian completion than the standard variants.

The formula for Turán number of spanning linear forests

Let F be a family of graphs. The Turán number ex(n;F) is defined to be the maximum number of edges in a graph of order n that is F-free. In 1959, Erdős and Gallai determined the Turán number of Mk+1 (a matching of size k+1) as follows: ex(n;Mk+1)=max2k+12,n2−n−k2.Since then, there has been a lot of research on Turán number of linear forests.A linear forest is a graph whose connected components are all paths or isolated vertices. Let Ln,k be the family of all linear forests of order n with k edges. In this paper, we prove that ex(n;Ln,k)=maxk2,n2−n−k−122+c,where c=0 if k is odd and c=1 otherwise. This determines the maximum number of edges in a non-Hamiltonian graph with given Hamiltonian completion number and also solves two open problems in Wang and Yang (2019) as special cases.Moreover, we show that our main theorem implies Erdős–Gallai Theorem and also gives a short new proof for it by the closure and counting techniques. Finally, we generalize our theorem to a conjecture which implies the famous Erdős Matching Conjecture.

Crossing-Optimal Acyclic HP-Completion for Outerplanar st-Digraphs

Given an embedded planar acyclic digraph G, we define the problem of acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM) to be the problem of determining a hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian acyclic digraph. Our results include: 1. We provide a characterization under which a planar st-digraph G is hamiltonian. 2. For an outerplanar st-digraph G, we define the st-polygon decomposition of G and, based on its properties, we develop a linear-time algorithm that solves the Acyclic-HPCCM problem. 3. For the class of planar st-digraphs, we establish an equivalence between the Acyclic-HPCCM problem and the problem of determining an upward 2-page topological book embedding with minimum number of spine crossings. We infer (based on this equivalence) for the class of outerplanar st-digraphs an upward topological 2-page book embedding with minimum number of spine crossings. To the best of our knowledge, it is the first time that edge-crossing minimization is studied in conjunction with the acyclic hamiltonian completion problem and the first time that an optimal algorithm with respect to spine crossing minimization is presented for upward topological book embeddings.

Local search algorithms for finding the Hamiltonian completion number of line graphs

Given a graph G=(V,E), the Hamiltonian completion number of G, HCN(G), is the minimum number of edges to be added to G to make it Hamiltonian. This problem is known to be \(\mathcal{NP}\) -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 of G. 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.

Hamiltonian completions of sparse random graphs

Given a (directed or undirected) graph G, finding the smallest number of additional edges which make the graph Hamiltonian is called the Hamiltonian Completion Problem (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.

Parallel algorithms for Hamiltonian problems on quasi-threshold graphs

In this paper we show structural and algorithmic properties on the class of quasi-threshold graphs, or QT-graphs for short, and prove necessary and sufficient conditions for a QT-graph to be Hamiltonian. Based on these properties and conditions, we construct an efficient parallel algorithm for finding a Hamiltonian cycle in a QT-graph; for an input graph on n vertices and m edges, our algorithm takes O( log n) time and requires O( n+ m) processors on the CREW PRAM model. In addition, we show that the problem of recognizing whether a QT-graph is a Hamiltonian graph and the problem of computing the Hamiltonian completion number of a nonHamiltonian QT-graph can also be solved in O( log n) time with O( n+ m) processors. Our algorithms rely on O( log n) -time parallel algorithms, which we develop here, for constructing tree representations of a QT-graph; we show that a QT-graph G has a unique tree representation, that is, a tree structure which meets the structural properties of G. We also present parallel algorithms for other optimization problems on QT-graphs which run in O( log n) time using a linear number of processors.

A linear algorithm for the Hamiltonian completion number of the line graph of a cactus

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.

Optimal Hamiltonian completions and path covers for trees, and a reduction to maximum flow

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.

Reduction Algorithms for Graphs of Small Treewidth

This paper presents a number of new ideas and results on graph reduction applied to graphs of bounded treewidth. S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese (J. Assoc. Comput. Mach.40, 1134–1164 (1993)) have shown that many decision problems on graphs can be solved in linear time on graphs of bounded treewidth, using a finite set of reduction rules. These algorithms can be used to solve problems on graphs of bounded treewidth without the need to obtain a tree decomposition of the input graph first. We show that the reduction method can be extended to solve the construction variants of many decision problems on graphs of bounded treewidth, including all problems definable in monadic second order logic. We also show that a variant of these reduction algorithms can be used to solve (constructive) optimization problems in O(n) time. For example, optimization and construction variants of INDEPENDENT SET and HAMILTONIAN COMPLETION NUMBER can be solved in this way on graphs of small treewidth. Additionally, we show that the results of H. L. Bodlaender and T. Hagerup (SIAM J. Comput.27, 1725–1746 (1998)) can be applied to our reduction algorithms, which results in parallel reduction algorithms that use O(n) operations and O(log n log* n) time on an EREW PRAM, or O(log n) time on a CRCW PRAM.

A linear algorithm for the Hamiltonian completion number of the line graph of a cactus

A linear algorithm for the Hamiltonian completion number of the line graph of a cactus

A linear algorithm for the Hamiltonian completion number of the line graph of a tree

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.

Hamiltonian powers in threshold and arborescent comparability graphs

We examine powers of Hamiltonian paths and cycles as well as Hamiltonian (power) completion problems in several highly structured graph classes. For threshold graphs we give efficient algorithms as well as sufficient and minimax toughness like conditions. For arborescent comparability graphs we have similar results but also show that for one type of completion problem an ·obvious’ minimax condition fails. For cographs we give examples showing that toughness and other ·obvious’ necessary conditions are not sufficient. For threshold graphs we give additional necessary and sufficient conditions in terms of vertex degrees as well as a minimax formula for the length of a longest cycle power.

Powers of Hamiltonian paths in interval graphs

We give a simple proof that the obvious necessary conditions for a graph to contain the kth power of a Hamiltonian path are sufficient for the class of interval graphs. The proof is based on showing that a greedy algorithm tests for the existence of Hamiltonian path powers in interval graphs. We will also discuss covers by powers of paths and analogues of the Hamiltonian completion number. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 31–38, 1998

The total interval number of a tree and the Hamiltonian completion number of its line graph

In this paper we investigate the problem of computing two parameters of a graph: the Total Interval Number TIN( G), and the Hamiltonian Completion Number HCN( G). We show that in case of a triangle-free graph these two parameters (TIN( G) and HCN( L( G)), where L( G) is the line graph of G) are related by a simple formula. We use this relationship to find a polynomial algorithm which determines TIN( G), for a tree G. Then we describe a construction which gives an interval representation of a tree G with TIN( G) number of intervals.

Combinatorial algorithms on a class of graphs

[From the original] For a class of graphs containing series–parallel and outerplanar graphs the linear-time solvability of a number of combinatorial problems such as minimum vertex cover, Hamiltonian completion problem, etc., is shown.

THE HAMILTONIAN COMPLETION PROBLEM AND ITS SOLUTION

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 linear algorithm for the hamiltonian completion number of a tree

A linear algorithm for the hamiltonian completion number of a tree

On the optional hamiltonian completion problem

AbstractThe Optional Hamiltonian Completion Problem is defined as follows: let the points V of a graph G be partitioned into a set V0 of optional points and a set V1 of non‐optional points; determine the minimum number of new lines which when added to G result in a graph which has a cycle containing every point of V1. This cycle may or may not contain optional points of V0. In this paper we present algorithms for solving this problem for trees, unicyclic graphs and cacti.