Hamiltonian Cycle Research Articles

RUTE TERPENDEK UNTUK PENGANGKUTAN SAMPAH DENGAN PENDEKATAN LINTASAN HAMILTON

This research is related to the route of picking up the waste which done by janitors in housing complex of Aur Duri Indah Rt.14 Jambi considering the condition of that housing which have some crossroads, such that janitors take the same road twice which seems inefficient in terms of time and fuel consumption. This research is aimed to identify scientifically about the shortest and more efficient route which should be taken by janitors. This problem is considered as optimization problem through graph modelling. To check the efficient route, the existence of Hamiltonian Path of the graph is identified by using Depth-First Search method, and so is the shortest path. It is found that there are 39 Hamiltonian paths and the shortest path has total distance of 793.8 meter. By the existence of the shortest Hamiltonian Path, it is confirmed that janitors in housing complex of Aur Duri Indah Rt.14 can travel each road once with total distance is about 793.8 meter.

Improving Multivariate Microaggregation through Hamiltonian Paths and Optimal Univariate Microaggregation

The collection of personal data is exponentially growing and, as a result, individual privacy is endangered accordingly. With the aim to lessen privacy risks whilst maintaining high degrees of data utility, a variety of techniques have been proposed, being microaggregation a very popular one. Microaggregation is a family of perturbation methods, in which its principle is to aggregate personal data records (i.e., microdata) in groups so as to preserve privacy through k-anonymity. The multivariate microaggregation problem is known to be NP-Hard; however, its univariate version could be optimally solved in polynomial time using the Hansen-Mukherjee (HM) algorithm. In this article, we propose a heuristic solution to the multivariate microaggregation problem inspired by the Traveling Salesman Problem (TSP) and the optimal univariate microaggregation solution. Given a multivariate dataset, first, we apply a TSP-tour construction heuristic to generate a Hamiltonian path through all dataset records. Next, we use the order provided by this Hamiltonian path (i.e., a given permutation of the records) as input to the Hansen-Mukherjee algorithm, virtually transforming it into a multivariate microaggregation solver we call Multivariate Hansen-Mukherjee (MHM). Our intuition is that good solutions to the TSP would yield Hamiltonian paths allowing the Hansen-Mukherjee algorithm to find good solutions to the multivariate microaggregation problem. We have tested our method with well-known benchmark datasets. Moreover, with the aim to show the usefulness of our approach to protecting location privacy, we have tested our solution with real-life trajectories datasets, too. We have compared the results of our algorithm with those of the best performing solutions, and we show that our proposal reduces the information loss resulting from the microaggregation. Overall, results suggest that transforming the multivariate microaggregation problem into its univariate counterpart by ordering microdata records with a proper Hamiltonian path and applying an optimal univariate solution leads to a reduction of the perturbation error whilst keeping the same privacy guarantees.

Random Perturbation of Sparse Graphs

In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Pósa and Korshunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.

What do Eulerian and Hamiltonian cycles have to do with genome assembly?

Many students are taught about genome assembly using the dichotomy between the complexity of finding Eulerian and Hamiltonian cycles (easy versus hard, respectively). This dichotomy is sometimes used to motivate the use of de Bruijn graphs in practice. In this paper, we explain that while de Bruijn graphs have indeed been very useful, the reason has nothing to do with the complexity of the Hamiltonian and Eulerian cycle problems. We give 2 arguments. The first is that a genome reconstruction is never unique and hence an algorithm for finding Eulerian or Hamiltonian cycles is not part of any assembly algorithm used in practice. The second is that even if an arbitrary genome reconstruction was desired, one could do so in linear time in both the Eulerian and Hamiltonian paradigms.

Network reliability in hamiltonian graphs

The reliability polynomial of a graph gives the probability that a graph remains operational when all its edges could fail independently with a certain fixed probability. In general, the problem of finding uniformly most reliable graphs inside a family of graphs, that is, one graph whose reliability is at least as large as any other graph inside the family, is very difficult. In this paper, we study this problem in the family of graphs containing a hamiltonian cycle.

Measuring what matters: A hybrid approach to dynamic programming with treewidth

We develop a framework for applying treewidth-based dynamic programming on graphs with “hybrid structure”, i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut.

A matheuristic algorithm for the pollution and energy minimization traveling salesman problems

AbstractThe pollution traveling salesman problem (PTSP) and the energy minimization traveling salesman problem (EMTSP) generalize the well‐known asymmetric traveling salesman problem by including environmental issues and the goal of reducing carbon emissions. Both problems call for determining a Hamiltonian tour that, in the PTSP, minimizes a function of fuel consumption and driver cost (where the fuel consumption depends on the distance traveled, the vehicle speed, and the vehicle load), while, in the EMTSP, minimizes a function depending on the vehicle load and the traveled distances. For both PTSP and EMTSP, we propose a matheuristic algorithm that uses the solution of the linear programming relaxation of a mixed integer linear programming model for the considered problem to determine good initial feasible solutions, applies a multioperator genetic algorithm to improve these solutions, and refines the best solution found through an iterated local search procedure. In order to evaluate the performance of the proposed matheuristics, we compare them with exact and heuristic algorithms from the literature on benchmark instances of both problems.

Mixed-integer programming approaches for the time-constrained maximal covering routing problem

In this paper, we study the recently introduced time-constrained maximal covering routing problem. In this problem, we are given a central depot, a set of facilities, and a set of customers. Each customer is associated with a subset of the facilities which can cover it. A feasible solution consists of k Hamiltonian cycles on subsets of the facilities and the central depot. Each cycle must contain the depot and must respect a given distance limit. The goal is to maximize the number of customers covered by facilities contained in the cycles. We develop two exact solution algorithms for the problem based on new mixed-integer programming models. One algorithm is based on a compact model, while the other model contains an exponential number of constraints, which are separated on-the-fly, i.e., we use branch-and-cut. We also describe preprocessing techniques, valid inequalities and primal heuristics for both models. We evaluate our solution approaches on the instances from literature and our algorithms are able to find the provably optimal solution for 267 out of 270 instances, including 123 instances, for which the optimal solution was not known before. Moreover, for most of the instances, our algorithms only take a few seconds, and thus are up to five magnitudes faster than previous approaches. Finally, we also discuss some issues with the instances from literature and present some new instances.

The solution of generalized stable sets and its refinement

In this paper, we review some results of the solution of generalized stable sets introduced by Van Deemen (1991) as a variant of stable sets for abstract decision problems. This solution will be investigated and reviewed for the more general case of irreflexive but not necessarily asymmetric or complete dominance relations. It is proven that the fundamental properties of this solution are preserved also for this kind of dominance relations. Two main shortcomings of the solution of generalized stable sets are firstly that it may contain Pareto-suboptimal alternatives when the dominance relation is derived from pairwise majority comparison, and secondly that it may fail to discriminate among the alternatives under consideration when the dominance relation is a Hamilton cycle, i.e. a cycle that includes all alternatives. A refinement of generalized stable sets is proposed in order to address these two shortcomings. This refinement is called the solution of undisturbed generalized stable sets. It will be shown that undisturbed generalized stable sets are Pareto optimal and have discriminating power in the case of Hamilton cycles. In addition, and perhaps more important, we prove that this refinement is always a subset of the solutions of the uncovered set and of the unsurpassed set. This implies that undisturbed generalized stable set also may be seen as a refinement of both the uncovered set and the unsurpassed set.

The threshold for the square of a Hamilton cycle

Resolving a conjecture of Kühn and Osthus from 2012, we show that p = 1 / n p= 1/\sqrt {n} is the threshold for the random graph G n , p G_{n,p} to contain the square of a Hamilton cycle.

Grouping memetic search for the colored traveling salesmen problem

The colored traveling salesmen problem is a node routing problem with multiple salesmen, where the cities are divided into m exclusive city sets and one shared city set. The objective is to minimize the total traveling distance of m Hamiltonian circuits (routes) under the following constraints: each exclusive city is to be visited by the corresponding salesman, while each shared city can be visited by any salesman. In this work, we present the first grouping memetic algorithm for solving this challenging problem. The algorithm includes three main components: (i) a greedy randomized heuristic for population initialization; (ii) a dedicated local search procedure for local optima exploration; (iii) a backbone-based crossover operator for solution recombination. We show computational results on three sets of 65 popular benchmark instances to demonstrate the competitiveness of our algorithm. We especially report improved upper bounds for 38 instances (for more than 58% cases). We also present first computational results with the general CPLEX solver, including 10 proven optimal solutions. Finally, we shed lights on the impacts of the key components of the algorithm. We make the code of the algorithm publicly available.

3D DNA Self-Assembly Algorithmic Model to Solve the Hamiltonian Path Problem

Self-assembly reveals the innate character of DNA computing, DNA self-assembly is regarded as the best way to make DNA computing transform into computer chip. This paper introduces a strategy of DNA 3D self-assembly algorithm to solve the Hamiltonian Path Problem. Firstly, I introduced a non-deterministic algorithm. Then, according to the algorithm I designed the types of DNA tiles which the computing process needs. Lastly, I demonstrated the self-assembly process and the experimental methods which can get the final result. The computing time is linear, and the number of the different tile types is constant.

Hamiltonian and pseudo-Hamiltonian cycles and fillings in simplicial complexes

We introduce and study a d-dimensional generalization of graph Hamiltonian cycles. These are the Hamiltonian d-dimensional cycles in Knd (the complete simplicial d-complex over a vertex set of size n). Hamiltonian d-cycles are the simple d-cycles of a complete rank, or, equivalently, of size 1+(n−1d).The discussion is restricted to the fields F2 and Q. For d=2, we characterize the n's for which Hamiltonian 2-cycles exist. For d=3 it is shown that Hamiltonian 3-cycles exist for infinitely many n's. In general, it is shown that there always exist simple d-cycles of size (n−1d)−O(nd−3). All the above results are constructive.Our approach naturally extends to (and in fact, involves) d-fillings, generalizing the notion of T-joins in graphs. Given a (d−1)-cycle Zd−1∈Knd, F is its d-filling if ∂F=Zd−1. We call a d-filling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size (n−1d). If a Hamiltonian d-cycle Z over F2 contains a d-simplex σ, then Z∖σ is a Hamiltonian d-filling of ∂σ (a closely related fact is also true for cycles over Q). Thus, the two notions are closely related.Most of the above results about Hamiltonian d-cycles hold for Hamiltonian d-fillings as well.

On Hamiltonian cycles in hypergraphs with dense link graphs

We show that every k-uniform hypergraph on n vertices whose minimum (k−2)-degree is at least (5/9+o(1))n2/2 contains a Hamiltonian cycle. A construction due to Han and Zhao shows that this minimum degree condition is optimal. The same result was proved independently by Lang and Sanhueza-Matamala.

Hamiltonicity of 3 t EC Graphs with α = κ + 1

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S . The minimum cardinality of a total dominating set of G is the total domination number γ t G of G . The graph G is total domination edge-critical, or γ t EC , if for every edge e in the complement of G , γ t G + e < γ t G . If G is γ t EC and γ t G = k , we say that G is k t EC . In this paper, we show that every 3 t EC graph with δ G ≥ 2 and α G = κ G + 1 has a Hamilton cycle.

POLYNOMIAL ALGORITHM FOR FINDING A HAMILTONIAN CYCLE ON A GRAPH

The subject of research is the solution of the problem of finding a Hamiltonian cycle on a graph, which in discrete mathematics belongs to the NP complexity class and still retains interest. The aim of this work is to develop a new algorithm for solving this problem, which guarantees finding the optimal solution with polynomial time. In [1], an analysis of the current state of the problem was performed, the shortcomings of existing methods of solving were noted, and new principles and method of finding solution were presented. Known methods for solving the problem are based on the implementation of a search of possible solutions or on intuitive heuristics. Enumeration methods of solutions are unacceptable in terms of costs, since they characterized by non-polynomial time. Heuristic methods are satisfactory in time, but they do not guarantee finding the optimal solution. The popularity of enumeration methods is explained by the linear simplicity of searching in a pre-known set of acceptable solutions to the problem. But the factorial dependence of the power of the set of solutions (n-1)! from the number of vertices of the graph n makes it impossible to use such methods for large-size problems in practice. The desire to significantly reduce time costs leads to attempts to reasonably reduce the enumeration set or to the development of various heuristics, which actually indicates that it is impossible to formulate conditions for finding the optimal solution to the problem as a whole. This article describes the conditions that determine the optimal solution of the problem and a polynomial algorithm for solving the problem that embodies the described method. Finding the optimal solution to the problem is reduced to finding a closed path in a new graph of shortest paths. The shortest paths graph built on the original graph of the problem by using Dijkstra's algorithm. The enumeration set for determining the optimal solution of the problem consists of solutions that are constructed from each vertex of the source graph in the shortest paths graph. The size of this set is estimated as n(n-1). The developed parallel algorithm guarantees finding the optimal solution, significantly reduces the initial search space, and allows you to find a solution with polynomial complexity. Testing of the program showed the efficiency of the developed method and algorithm for solving the problem.

A Novel Interconnection Network with Improved Network Cost Through Shuffle-Exchange Permutation Graph

The interconnection network represents an interconnected structure of processors that strongly determines the performance quality of a parallel processing system. The shuffle-exchange permutation (SEP) network with three degrees has high fault tolerance and can be efficiently simulated through star, bubble-sort, and pancake graphs. This study proposes a new interconnection network: the new SEP (NSEP), which improves the diameter and reduces network cost by adding one edge to the SEP network, and presents its graph properties and routing algorithms. The NSEP network, with a degree of connectivity of four, demonstrated maximum fault tolerance and Hamiltonian cycle. Furthermore, the diameter was seen to be improved by 40% or more and the network cost by 20% or more.

Hyper-Hamiltonian circulants

A Hamiltonian graph G = ( V,E ) is called hyper-Hamiltonian if G - v is Hamiltonian for any v ∈ V ( G ). G is called a circulant if its automorphism group contains a | V ( G )|-cycle. First, we give the necessary and sufficient conditions for any undirected connected circulant to be hyper-Hamiltonian. Second, we give necessary and sufficient conditions for a connected circulant digraph with two jumps to be hyper-Hamiltonian. In addition, we specify some sufficient conditions for a circulant digraph with arbitrary number of jumps to be hyper-Hamiltonian.

Enumeration of Labeled and Unlabeled Hamiltonian Cycles in Complete k-Partite Graphs

Labeled and unlabeled Hamiltonian cycles in complete n-partite graphs Kd,d,...,d having exactly d vertices in each part are enumerated. Recurrence relations for the exact values $$ {b}_n^{(d)} $$ of such cycles for arbitrary n and d are obtained.

Analyzing blockchain adoption barriers in manufacturing supply chains by the neutrosophic analytic hierarchy process

Tools established for managing information flow in supply chain management and logistics should match digital transformations. This issue is particularly salient for developing nations that hope to achieve sustainable development goals in a globalized era. Modern technologies are required to ensure a secure, transparent, and traceable path of information flow in global supply chains; however, it is not always straightforward for businesses in developing economies to adopt new digital technologies while sustaining productivity. One of the foundational technologies that can be used to create a basis for economic and social systems and to affect manufacturing supply chains in developing economies is blockchain. In this study, we analyze the barriers to blockchain technology adoption in manufacturing supply chains using the neutrosophic analytic hierarchy process (N-AHP). We propose an action plan framework for the validation of blockchain technology in a developing economy. The findings demonstrate that “transaction-level uncertainties” comprise the most critical barrier and have the highest weight in the final ranking followed by “usage in the underground economy”, “managerial commitment”, “challenges in scalability”, and “privacy risks”. This paper can assist industrial managers and experts in emerging economies to more clearly identify barriers to the implementation of blockchain technology and show them how to successfully employ blockchain technology in their supply chains.