Hamiltonian Path Problem Research Articles

The Hardest Hamiltonian Cycle Problem Instances: The Plateau of Yes and the Cliff of No

We use two evolutionary algorithms to make hard instances of the Hamiltonian cycle problem. Hardness (or ‘fitness’), is defined as the number of recursions required by Vandegriend–Culberson, the best known exact backtracking algorithm for the problem. The hardest instances, all non-Hamiltonian, display a high degree of regularity and scalability across graph sizes. These graphs are found multiple times through independent runs, and by both evolutionary algorithms, suggesting the search space might contain monotonic paths towards the global maximum. For Hamiltonian-bound evolution, some hard graphs were found, but convergence is much less consistent. In this extended paper, we survey the neighbourhoods of both the hardest yes- and no-instances produced by the evolutionary algorithms. Results show that the hardest no-instance resides on top of a steep cliff, while the hardest yes-instance turns out to be part of a plateau of 27 equally hard instances. While definitive answers are far away, the results provide a lot of insight in the Hamiltonian cycle problem’s state space.

A quantum approximate optimization algorithm for solving Hamilton path problem

A quantum approximate optimization algorithm for solving Hamilton path problem

Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems

This paper studies the Hamiltonian cycle problem (HCP) and the traveling salesman problem (TSP) on D-Wave quantum systems. Motivated by the fact that most libraries present their benchmark instances in terms of adjacency matrices, we develop a novel matrix formulation for the HCP and TSP Hamiltonians, which enables the seamless and automatic integration of benchmark instances in quantum platforms. We also present a thorough mathematical analysis of the precise number of constraints required to express the HCP and TSP Hamiltonians. This analysis explains quantitatively why, almost always, running incomplete graph instances requires more qubits than complete instances. It turns out that QUBO models for incomplete graphs require more quadratic constraints than complete graphs, a fact that has been corroborated by a series of experiments. Moreover, we introduce a technique for the min-max normalization for the coefficients of the TSP Hamiltonian to address the problem of invalid solutions produced by the quantum annealer, a trend often observed. Our extensive experimental tests have demonstrated that the D-Wave Advantage_system4.1 is more efficient than the Advantage_system1.1, both in terms of qubit utilization and the quality of solutions. Finally, we experimentally establish that the D-Wave hybrid solvers always provide valid solutions, without violating the given constraints, even for arbitrarily big problems up to 120 nodes.

Hamiltonian problems in directed graphs with simple row patterns

We define a row pattern of a matrix as a model (or a condition) to be satisfied by each row of the matrix, independently of the other rows. When applied to adjacency matrices of digraphs, a row pattern allows to define a class of digraphs characterized by a local condition on the out-neighborhood of each vertex, once an appropriate linear ordering of the vertices has been found.We show that the Hamiltonian Cycle and Hamiltonian Path problems are NP-complete on digraphs with simple row patterns, namely C1P-digraphs (where the row pattern describes the consecutive ones property), 2-out-regular digraphs (where each row contains exactly two values 1) and sym-digraphs (where the 1s in each row are symmetric with respect to the main diagonal, as soon as the row contains at least two 1s).Classes with simple row patterns thus show a real structural complexity, despite their simple definitions. Moreover, they raise interesting class recognition problems.

Cyclability in graph classes

A subset T⊆V(G) of vertices of a graph G is said to be cyclable if G has a cycle C containing every vertex of T, and for a positive integer k, a graph G is k-cyclable if every set of vertices of size at most k is cyclable. The Terminal Cyclability problem asks, given a graph G and a set T of vertices, whether T is cyclable, and the k-Cyclability problem asks, given a graph G and a positive integer k, whether G is k-cyclable. These problems are generalizations of the classical Hamiltonian Cycle problem. We initiate the study of these problems for graph classes that admit polynomial algorithms for Hamiltonian Cycle. We show that Terminal Cyclability can be solved in linear time for interval graphs, bipartite permutation graphs and cographs. Moreover, we construct certifying algorithms that either produce a solution, that is a cycle, or output a graph separator that certifies a no-answer. We use these results to show that k-Cyclability can be solved in polynomial time when restricted to the aforementioned graph classes.

Finding Hamiltonian and Longest (s,t)-Paths of C-Shaped Supergrid Graphs in Linear Time

A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

Solving the multi-objective Hamiltonian cycle problem using a Branch-and-Fix based algorithm

The Hamiltonian cycle problem consists of finding a cycle in a given graph that passes through every single vertex exactly once, or determining that this cannot be achieved. In this investigation, a graph is considered with an associated set of matrices. The entries of each of the matrix correspond to a different weight of an arc. A multi-objective Hamiltonian cycle problem is addressed here by computing a Pareto set of solutions that minimize the sum of the weights of the arcs for each objective. Our heuristic approach extends the Branch-and-Fix algorithm, an exact method that embeds the problem in a stochastic process. To measure the efficiency of the proposed algorithm, we compare it with a multi-objective genetic algorithm in graphs of a different number of vertices and density. The results show that the density of the graphs is critical when solving the problem. The multi-objective genetic algorithm performs better (quality of the Pareto sets) than the proposed approach in random graphs with high density; however, in these graphs it is easier to find Hamiltonian cycles, and they are closer to the multi-objective traveling salesman problem. The results reveal that, in a challenging benchmark of Hamiltonian graphs with low density, the proposed approach significantly outperforms the multi-objective genetic algorithm.

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.

Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

Symmetric Connectivity of Underwater Acoustic Sensor Networks Based on Multi-Modal Directional Transducer.

Topology control is one of the most essential technologies in wireless sensor networks (WSNs); it constructs networks with certain characteristics through the usage of some approaches, such as power control and channel assignment, thereby reducing the inter-nodes interference and the energy consumption of the network. It is closely related to the efficiency of upper layer protocols, especially MAC and routing protocols, which are the same as underwater acoustic sensor networks (UASNs). Directional antenna technology (directional transducer in UASNs) has great advantages in minimizing interference and conserving energy by restraining the beamforming range. It enables nodes to communicate with only intended neighbors; nevertheless, additional problems emerge, such as how to guarantee the connectivity of the network. This paper focuses on the connectivity problem of UASNs equipped with tri-modal directional transducers, where the orientation of a transducer is stabilized after the network is set up. To efficiently minimize the total network energy consumption under constraint of connectivity, the problem is formulated to a minimum network cost transducer orientation (MNCTO) problem and is provided a reduction from the Hamiltonian path problem in hexagonal grid graphs (HPHGG), which is proved to be NP-complete. Furthermore, a heuristic greedy algorithm is proposed for MNCTO. The simulation evaluation results in a contrast with its omni-mode peer, showing that the proposed algorithm greatly reduces the network energy consumption by up to nearly half on the premise of satisfying connectivity.

Computing directed Steiner path covers

In this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph G=(V,E) and a set T subseteq V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

Approximation algorithms for the k-depots Hamiltonian path problem

Approximation algorithms for the k-depots Hamiltonian path problem

The Steiner cycle and path cover problem on interval graphs

The Steiner path problem is a common generalization of the Steiner tree and the Hamiltonian path problem, in which we have to decide if for a given graph there exists a path visiting a fixed set of terminals. In the Steiner cycle problem we look for a cycle visiting all terminals instead of a path. The Steiner path cover problem is an optimization variant of the Steiner path problem generalizing the path cover problem, in which one has to cover all terminals with a minimum number of paths. We study those problems for the special class of interval graphs. We present linear time algorithms for both the Steiner path cover problem and the Steiner cycle problem on interval graphs given as endpoint sorted lists. The main contribution is a lemma showing that backward steps to non-Steiner intervals are never necessary. Furthermore, we show how to integrate this modification to the deferred-query technique of Chang et al. to obtain the linear running times.

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.

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.

Application of DNA Nanoparticle Conjugation on the Hamiltonian Path Problem

A DNA computing algorithm is proposed in this paper. The algorithm uses the assembly of DNA/Au nanoparticle conjugation to solve an NP-complete problem in the Graph theory, the Hamiltonian Path problem. According to the algorithm, I designed the special DNA/Au nanoparticle conjugations which assembled based on a specific graph, then, a series of experimental techniques are utilized to get the final result. This biochemical algorithm can reduce the complexity of the Hamiltonian Path problem greatly, which will provide a practical way to the best use of DNA self-assembly model.

Feasible Bases for a Polytope Related to the Hamilton Cycle Problem

We study a certain polytope depending on a graph G and a parameter β ∈ (0,1) that arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Literature suggests a conjecture a lower bound on the proportion of feasible bases corresponding to Hamiltonian cycles in the set of all feasible bases. We make progress toward a proof of the conjecture by proving results about the structure of feasible bases. In particular, we prove three main results: (1) the set of feasible bases is independent of the parameter β when the parameter is close to one, (2) the polytope can be interpreted as a generalized network flow polytope, and (3) we deduce a combinatorial interpretation of the feasible bases. We also provide a full characterization for a special class of feasible bases, and we apply this to provide some computational support for the conjecture.

A synthetic biology approach for the design of genetic algorithms with bacterial agents

ABSTRACT Bacteria have been a source of inspiration for the design of evolutionary algorithms. At the beginning of the twentieth century synthetic biology was born, a discipline whose goal is the design of biological systems that do not exist in nature, for example, programmable synthetic bacteria. In this paper, we introduce as a novelty the designing of evolutionary algorithms where all the steps are conducted by synthetic bacteria. To this end, we designed a genetic algorithm, which we have named BAGA, illustrating its utility solving simple instances of optimisation problems such as function optimisation, 0/1 knapsack problem, Hamiltonian path problem. The results obtained open the possibility of conceiving evolutionary algorithms inspired by principles, mechanisms and genetic circuits from synthetic biology. In summary, we can conclude that synthetic biology is a source of inspiration either for the design of evolutionary algorithms or for some of their steps, as shown by the results obtained in our simulation experiments.

Nanopore decoding for a Hamiltonian path problem

DNA computing has attracted attention as a tool for solving mathematical problems due to the potential for massive parallelism with low energy consumption. However, decoding the output information to a human-recognizable signal is generally time-consuming owing to the requirement for multiple steps of biological operations. Here, we describe simple and rapid decoding of the DNA-computed output for a directed Hamiltonian path problem (HPP) using nanopore technology. In this approach, the output DNA duplex undergoes unzipping whilst passing through an α-hemolysin nanopore, with information electrically decoded as the unzipping time of the hybridized strands. As a proof of concept, we demonstrate nanopore decoding of the HPP of a small graph encoded in DNA. Our results show the feasibility of nanopore measurement as a rapid and label-free decoding method for mathematical DNA computation using parallel self-assembly.

Kernelization of Graph Hamiltonicity: Proper $H$-Graphs

We obtain new polynomial kernels and compression algorithms for Path Cover and Cycle Cover, the well-known generalizations of the classical Hamiltonian Path and Hamiltonian Cycle problems. Our choi...