Hamiltonian Path Problem Research Articles

A linear-time algorithm for finding Hamiltonian (s,t)-paths in even-sized rectangular grid graphs with a rectangular hole

The Hamiltonian path problem for general grid graphs is NP-complete. In this paper, we give the necessary conditions for the existence of a Hamiltonian path between two given vertices in a rectangular grid graph with a rectangular hole; where the size of graph is even. In addition, we show that the Hamiltonian path in these graphs can be computed in linear-time.

Time complexity of two disjoint simple paths

Finding two disjoint simple paths on two given sets of points is a geometric problem introduced by Jeff Erickson. This problem has various applications in computational geometry, like robot motion planning, generating polygon etc. We will present a reduction from planar Hamiltonian path to this problem, and prove that it is NP-Complete. To the best of our knowledge, no study has considered its complexity up until now. We also present a reduction from planar Hamiltonian path problem to the problem of finding a path on given points in the presence of arbitrary obstacles and prove that it is NP-Complete too. Also, we present a heuristic algorithm with time complexity of O(n4) to solve this problem. The proposed algorithm first calculates the convex hull for each of the entry points and then produces two simple paths on the two entry point sets

A heuristic for a heterogeneous automated guided vehicle routing problem

The article addresses a Heterogeneous Automated Guided Vehicle (AGV) Routing Problem that distributes given jobs to one of the AGVs and finds a route for each AGV while minimizing the sum of tour costs. AGVs are heterogeneous as the cost of traveling between any two locations varies depending on the type of AGV. The problem is a generalization of multiple heterogeneous asymmetric Hamiltonian path problem and a primal-dual heuristic has been developed. Despite the complexity of the problem, the computational results show that the proposed algorithm produces good quality solutions within a relatively short computation time.

Parameterized Edge Hamiltonicity

We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixed-parameter tractability results. First, we settle an open question of Demaine et al. (2014) by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixed-parameter tractability even for a generalization of the problem to arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set. As an interesting consequence, we show that this implies an FPT algorithm for (Vertex) Hamiltonian Path parameterized by (vertex) clique cover. We also consider the problem parameterized by treewidth or clique-width. Surprisingly, we show that the problem is FPT for both of these standard parameters, in contrast to its vertex version, which is W[1]-hard for clique-width. Our technique, which may be of independent interest, relies on a structural characterization of clique-width in terms of treewidth and complete bipartite subgraphs due to Gurski and Wanke.

A linear-time algorithm for finding Hamiltonian (s, t)-paths in odd-sized rectangular grid graphs with a rectangular hole

A grid graph $$G_{\mathrm{g}}$$ is a finite vertex-induced subgraph of the two-dimensional integer grid $$G^\infty $$ . A rectangular grid graph R(m, n) is a grid graph with horizontal size m and vertical size n. A rectangular grid graph with a rectangular hole is a rectangular grid graph R(m, n) such that a rectangular grid subgraph R(k, l) is removed from it. The Hamiltonian path problem for general grid graphs is NP-complete. In this paper, we give necessary conditions for the existence of a Hamiltonian path between two given vertices in an odd-sized rectangular grid graph with a rectangular hole. In addition, we show that how such paths can be computed in linear time.

Discrete Differential Geometry and the Structural Study of Protein Complexes

This paper proposes a novel four-dimensional approach to the structural study of protein complexes. In the approach, the surface of a protein molecule is to be described using the intersection of a pair of four-dimensional triangular cones (with multiple top vertexes). As a mathematical toy model of protein complexes, we consider complexes of closed trajectories of n-simplices (n=2,3,4...), where the design problem of protein complexes corresponds to an extended version of the Hamiltonian cycle problem. The problem is to find “a set of” closed trajectories of n-simplices which fills the n-dimensional region defined by a given pair of n+1 -dimensional triangular cones. Here we give a solution to the extended Hamiltonian cycle problem in the case of n=2 using the discrete differential geometry of triangles (i.e., 2-simplices).

Generative capacity of sticker systems with the presence of weights

DNA computing involves computing models which use the recombination behavior of DNA molecules as computation devices. This idea was successfully applied by Adleman in his biological experiment in order to show the solvability of the Hamiltonian path problem for larger instances. In 1998, a DNA based computation model, called sticker system, which is an abstraction of the computations using the recombination behavior as in Adleman’s experiment was proposed. In 2013, a new variant of sticker system using weights was introduced, namely the weighted sticker system. In this paper, the generative capacity for several variants of bounded delay and nonrestrictive weighted sticker systems are investigated. The relation between families of languages generated by several variants of weighted sticker systems and weighted grammars are also presented.

MREA: a minimum resource expenditure node capture attack in wireless sensor networks

AbstractBecause of the stochastic key pre‐distribution and complicated network topology, designing an energy‐efficient node capture attack algorithm is of great challenge. Although many algorithms have been proposed for node capture attack, previous methods lack of concerning minimizing resource expenditure in modeling attacking behavior. In this paper, we propose a novel way of modeling the node capture attack. First, we transform the problem into a set covering problem with a shortest Hamiltonian cycle problem, which has been shown to be NP‐hard. Consequently, we also develop a heuristic called minimum resource expenditure node capture attack (MREA) to maximize destructiveness while minimizing resource expenditure. Moreover, extensive simulations are conducted to show the performance of MREA. Simulation results show that MREA outperforms other algorithms in reducing the attack rounds and saving resource expenditure. Copyright © 2016 John Wiley & Sons, Ltd.

Reducing the generalised Sudoku problem to the Hamiltonian cycle problem

The generalised Sudoku problem with N symbols is known to be NP-complete, and hence is equivalent to any other NP-complete problem, even for the standard restricted version where N is a perfect square. In particular, generalised Sudoku is equivalent to the, classical, Hamiltonian cycle problem. A constructive algorithm is given that reduces generalised Sudoku to the Hamiltonian cycle problem, where the resultant instance of Hamiltonian cycle problem is sparse, and has O(N3) vertices. The Hamiltonian cycle problem instance so constructed is a directed graph, and so a (known) conversion to undirected Hamiltonian cycle problem is also provided so that it can be submitted to the best heuristics. A simple algorithm for obtaining the valid Sudoku solution from the Hamiltonian cycle is provided. Techniques to reduce the size of the resultant graph are also discussed.

Molecular solutions for the Maximum K-colourable Sub graph Problem in Adleman-Lipton model

Adleman (1) showed that deoxyribonucleic acid (DNA) strands could be employed towards calculating solutions to an instance of the Hamiltonian path problem (HPP). Lipton (5) also demonstrated that Adleman's techniques could be used to solve the Satisfiability problem. In this paper, we use Adleman-Lipton model for developing a DNA algorithm to solve Maximum k-colourable Sub graph problem. In spite of the NP-hardness of Maximum k-colourable Sub graph problem our DNA procedures is done in a polynomial time. Recently, DNA computing has considerable attention as one of non-silicon based computing. Watson-Crick complementarity and massive parallelism are two important features of DNA. By using these features, one can solve an NP-complete problem, which usually needs exponential time on a silicon-based computer, in a polynomial number of steps with DNA molecules (3). Adleman (1) solved Hamiltonian path problem of size n in spite of NP-hardness of the problem in O(n) steps using DNA molecules. That is the first work for DNA computing. The second NP-hard problem that has solved by DNA computing is Satisfiability (SAT), Lipton (5) showed that the Adelman's manner could be used to determine SAT. Moreover, procedures for primitive operations, such as logic or arithmetic operations, have also been proposed so as to apply DNA computing in a wide range of problems (3-4, 6-14). In this paper, the DNA operations proposed by Adleman (1) and Lipton (5) are used for figuring out solutions of Maximum k- colourable Sub graph problem. Given an undirected graph G = (V, E) with an assignment of weights to the edges w: E → N and an integer k ∈ {2, 3, …, |V|}, we try to find maximum

Efficient solution for finding Hamilton cycles in undirected graphs.

The Hamilton cycle problem is closely related to a series of famous problems and puzzles (traveling salesman problem, Icosian game) and, due to the fact that it is NP-complete, it was extensively studied with different algorithms to solve it. The most efficient algorithm is not known. In this paper, a necessary condition for an arbitrary un-directed graph to have Hamilton cycle is proposed. Based on this condition, a mathematical solution for this problem is developed and several proofs and an algorithmic approach are introduced. The algorithm is successfully implemented on many Hamiltonian and non-Hamiltonian graphs. This provides a new effective approach to solve a problem that is fundamental in graph theory and can influence the manner in which the existing applications are used and improved.

Planarizing Gadgets for Perfect Matching Do Not Exist

To reduce a graph problem to its planar version, a standard technique is to replace crossings in a drawing of the input graph by planarizing gadgets. We show unconditionally that such a reduction is not possible for the perfect matching problem and also extend this to some other problems related to perfect matching. We further show that there is no planarizing gadget for the Hamiltonian cycle problem.

Faster than classical quantum algorithm for dense formulas of exact satisfiability and occupation problems

We present an exact quantum algorithm for solving the Exact Satisfiability problem, which belongs to the important NP-complete complexity class. The algorithm is based on an intuitive approach that can be divided into two parts: the first step consists in the identification and efficient characterization of a restricted subspace that contains all the valid assignments of the Exact Satisfiability; while the second part performs a quantum search in such restricted subspace. The quantum algorithm can be used either to find a valid assignment (or to certify that no solution exists) or to count the total number of valid assignments. The query complexities for the worst-case are respectively bounded by and , where n is the number of variables and the number of linearly independent clauses. Remarkably, the proposed quantum algorithm results to be faster than any known exact classical algorithm to solve dense formulas of Exact Satisfiability. As a concrete application, we provide the worst-case complexity for the Hamiltonian cycle problem obtained after mapping it to a suitable Occupation problem. Specifically, we show that the time complexity for the proposed quantum algorithm is bounded by for 3-regular undirected graphs, where n is the number of nodes. The same worst-case complexity holds for -regular bipartite graphs. As a reference, the current best classical algorithm has a (worst-case) running time bounded by . Finally, when compared to heuristic techniques for Exact Satisfiability problems, the proposed quantum algorithm is faster than the classical WalkSAT and Adiabatic Quantum Optimization for random instances with a density of constraints close to the satisfiability threshold, the regime in which instances are typically the hardest to solve. The proposed quantum algorithm can be straightforwardly extended to the generalized version of the Exact Satisfiability known as Occupation problem. The general version of the algorithm is presented and analyzed.

A P system for Hamiltonian cycle problem

P system is a new kind of distributed parallel computing model. In the P system, objects in each membrane can follow the evolution of the maximum parallelism principle, so we can solve NP-hard problem in polynomial time. In this paper, we design a family of P system with to judge whether there is a Hamiltonian cycle and find all Hamiltonian cycle in an undirected graph, and then an instance is given to illustrate the feasibility and effectiveness of our designed P systems.

Hamiltonian cycles in linear-convex supergrid graphs

A supergrid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional supergrid. The supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. The Hamiltonian cycle problem for grid and triangular grid graphs was known to be NP-complete. Recently, we have proved the Hamiltonian cycle problem on supergrid graphs to be NP-complete. The Hamiltonian cycle problem on supergrid graphs can be applied to control the stitching trace of computerized sewing machines. In this paper, we will study the Hamiltonian cycle property of linear-convex supergrid graphs which form a subclass of supergrid graphs. A connected graph is called k-connected if there are k vertex-disjoint paths between every pair of vertices, and is called locally connected if the neighbors of each vertex in it form a connected subgraph. In this paper, we first show that any 2-connected, linear-convex supergrid graph is locally connected. We then give constructive proofs to show that any 2-connected, linear-convex supergrid graph contains a Hamiltonian cycle. Based on the constructive proofs, we finally present a linear-time algorithm to construct a Hamiltonian cycle of a 2-connected, linear-convex supergrid graph.

Constructing arbitrarily large graphs with a specified number of Hamiltonian cycles

A constructive method is provided that outputs a directed graph which is named a broken crown graph, containing $5n-9$ vertices and $k$ Hamiltonian cycles for any choice of integers $n \geq k \geq 4$. The construction is not designed to be minimal in any sense, but rather to ensure that the graphs produced remain non-trivial instances of the Hamiltonian cycle problem even when $k$ is chosen to be much smaller than $n$.

Global cycle properties of locally isometric graphs

Let P be a graph property. A graph G is said to be locallyP if the subgraph induced by the open neighbourhood of every vertex in G has property P. Ryjáček’s well-known conjecture that every connected, locally connected graph is weakly pancyclic motivated us to consider the global cycle structure of locally P graphs, where P is the property of having diameter at most k for some fixed k≥1. For k=2 these graphs are called locally isometric graphs. For Δ≤5, we obtain a complete structural characterization of locally isometric graphs that are not fully cycle extendable. For Δ=6, it is shown that locally isometric graphs that are not fully cycle extendable contain a pair of true twins of degree 6. Infinite classes of locally isometric graphs with Δ=6 that are not fully cycle extendable are described and observations are made that suggest that a complete characterization of these graphs is unlikely. It is shown that Ryjáček’s conjecture holds for all locally isometric graphs with Δ≤6. The Hamilton cycle problem for locally isometric graphs with maximum degree at most 8 is shown to be NP-complete.

Hamiltonian paths in L-shaped grid graphs

Grid graphs are subgraphs of the infinite 2-dimensional integer grid. The Hamiltonian path problem for general grid graphs is a well-known NP-complete problem. In this paper, we present necessary and sufficient conditions for the existence of a Hamiltonian path between two given vertices in L-shaped grid graphs. We also show that a Hamiltonian path between two given vertices of a L-shaped grid graph can be computed in linear time.

Algebraic Dynamic Programming over general data structures.

BackgroundDynamic programming algorithms provide exact solutions to many problems in computational biology, such as sequence alignment, RNA folding, hidden Markov models (HMMs), and scoring of phylogenetic trees. Structurally analogous algorithms compute optimal solutions, evaluate score distributions, and perform stochastic sampling. This is explained in the theory of Algebraic Dynamic Programming (ADP) by a strict separation of state space traversal (usually represented by a context free grammar), scoring (encoded as an algebra), and choice rule. A key ingredient in this theory is the use of yield parsers that operate on the ordered input data structure, usually strings or ordered trees. The computation of ensemble properties, such as a posteriori probabilities of HMMs or partition functions in RNA folding, requires the combination of two distinct, but intimately related algorithms, known as the inside and the outside recursion. Only the inside recursions are covered by the classical ADP theory.ResultsThe ideas of ADP are generalized to a much wider scope of data structures by relaxing the concept of parsing. This allows us to formalize the conceptual complementarity of inside and outside variables in a natural way. We demonstrate that outside recursions are generically derivable from inside decomposition schemes. In addition to rephrasing the well-known algorithms for HMMs, pairwise sequence alignment, and RNA folding we show how the TSP and the shortest Hamiltonian path problem can be implemented efficiently in the extended ADP framework. As a showcase application we investigate the ancient evolution of HOX gene clusters in terms of shortest Hamiltonian paths.ConclusionsThe generalized ADP framework presented here greatly facilitates the development and implementation of dynamic programming algorithms for a wide spectrum of applications.

Torus–Connected Cycles: A Simple and Scalable Topology for Interconnection Networks

Abstract Supercomputers are today made up of hundreds of thousands of nodes. The interconnection network is responsible for connecting all these nodes to each other. Different interconnection networks have been proposed; high performance topologies have been introduced as a replacement for the conventional topologies of recent decades. A high order, a low degree and a small diameter are the usual properties aimed for by such topologies. However, this is not sufficient to lead to actual hardware implementations. Network scalability and topology simplicity are two critical parameters, and they are two of the reasons why modern supercomputers are often based on torus interconnection networks (e.g., Fujitsu K, IBM Sequoia). In this paper we first describe a new topology, torus-connected cycles (TCCs), realizing a combination of a torus and a ring, thus retaining interesting properties of torus networks in addition to those of hierarchical interconnection networks (HINs). Then, we formally establish the diameter of a TCC, and deduce a point-to-point routing algorithm. Next, we propose routing algorithms solving the Hamiltonian cycle problem, and, in a two dimensional TCC, the Hamiltonian path one. Correctness and complexities are formally proved. The proposed algorithms are time-optimal.