Hamiltonian Path Problem Research Articles

Not being (super)thin or solid is hard: A study of grid Hamiltonicity

We give a systematic study of Hamiltonicity of grids — the graphs induced by finite subsets of vertices of the tilings of the plane with congruent regular convex polygons (triangles, squares, or hexagons). Summarizing and extending existing classification of the usual, “square”, grids, we give a comprehensive taxonomy of the grid graphs. For many classes of grid graphs we resolve the computational complexity of the Hamiltonian cycle problem. For graphs for which there exists a polynomial-time algorithm we give efficient algorithms to find a Hamiltonian cycle. We also establish, for any g ⩾ 6 , a one-to-one correspondence between Hamiltonian cycles in planar bipartite maximum-degree-3 graphs and Hamiltonian cycles in the class C g of girth-g planar maximum-degree-3 graphs. As applications of the correspondence, we show that for graphs in C g the Hamiltonian cycle problem is NP-complete and that for any N ⩾ 5 there exist graphs in C g that have exactly N Hamiltonian cycles. We also prove that for the graphs in C g , a Chinese Postman tour gives a ( 1 + 8 g ) -approximation to TSP, improving thereby the Christofides ratio when g > 16 . We show further that, in any graph, the tour obtained by Christofides' algorithm is not longer than a Chinese Postman tour.

Laminar structure of ptolemaic graphs with applications

Ptolemaic graphs are those satisfying the Ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs. It can also be seen as a natural generalization of block graphs (and hence trees). In this paper, we first state a laminar structure of cliques, which leads to its canonical tree representation. This result is a translation of γ -acyclicity which appears in the context of relational database schemes. The tree representation gives a simple intersection model of ptolemaic graphs, and it is constructed in linear time from a perfect elimination ordering obtained by the lexicographic breadth first search. Hence the recognition and the graph isomorphism for ptolemaic graphs can be solved in linear time. Using the tree representation, we also give an efficient algorithm for the Hamiltonian cycle problem.

On the Hamiltonicity Gap and doubly stochastic matrices

AbstractWe consider the Hamiltonian cycle problem embedded in singularly perturbed (controlled) Markov chains. We also consider a functional on the space of stationary policies of the process that consists of the (1,1)‐entry of the fundamental matrices of the Markov chains induced by these policies. We focus on the subset of these policies that induce doubly stochastic probability transition matrices which we refer to as the “doubly stochastic policies.” We show that when the perturbation parameter, ε, is sufficiently small, the minimum of this functional over the space of the doubly stochastic policies is attained at a Hamiltonian cycle, provided that the graph is Hamiltonian. We also show that when the graph is non‐Hamiltonian, the above minimum is strictly greater than that in a Hamiltonian case. We call the size of this difference the “Hamiltonicity Gap” and derive a conservative lower bound for this gap. Our results imply that the Hamiltonian cycle problem is equivalent to the problem of minimizing the variance of the first hitting time of the home node, over doubly stochastic policies. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

The Linear Ordering Problem with cumulative costs

The optimization problem of finding a permutation of a given set of items that minimizes a certain cost function is naturally modeled by introducing a complete digraph G whose vertices correspond to the items to be sorted. Depending on the cost function to be used, different optimization problems can be defined on G. The most familiar one is the min-cost Hamiltonian path problem (or its closed-path version, the Traveling Salesman Problem), arising when the cost of a given permutation only depends on consecutive node pairs. A more complex situation arises when a given cost has to be paid whenever an item is ranked before another one in the final permutation. In this case, a feasible solution is associated with an acyclic tournament (the transitive closure of an Hamiltonian path), and the resulting problem is known as the Linear Ordering Problem (LOP). In this paper we introduce and study a relevant case of LOP arising when the overall permutation cost can be expressed as the sum of terms α u associated with each item u, each defined as a linear combination of the values α v of all items v that follow u in the permutation. This setting implies a cumulative (nonlinear) propagation of the value of variables α v along the node permutation, hence the name Linear Ordering Problem with Cumulative Costs. We illustrate the practical application in wireless telecommunication system that motivated the present study. We prove complexity results, and propose a Mixed-Integer Linear Programming model as well as an ad hoc enumerative algorithm for the exact solution of the problem. A dynamic-programming heuristic is also described. Extensive computational results on large sets of instances are presented, showing that the proposed techniques are capable of solving, in reasonable computing times, all the instances coming from our application.

A new formulation for the Traveling Deliveryman Problem

The Traveling Deliveryman Problem is a generalization of the Minimum Cost Hamiltonian Path Problem where the starting vertex of the path, i.e. a depot vertex, is fixed in advance and the cost associated with a Hamiltonian path equals the sum of the costs for the layers of paths (along the Hamiltonian path) going from the depot vertex to each of the remaining vertices. In this paper, we propose a new Integer Programming formulation for the problem and computationally evaluate the strength of its Linear Programming relaxation. Computational results are also presented for a cutting plane algorithm that uses a number of valid inequalities associated with the proposed formulation. Some of these inequalities are shown to be facet defining for the convex hull of feasible solutions to that formulation. These inequalities proved very effective when used to reinforce Linear Programming relaxation bounds, at the nodes of a Branch and Bound enumeration tree.

Optimal wire ordering and spacing in low power semiconductor design

A key issue for high integration circuit design in the semiconductor industry are power constraints that stem from the need for heat removal and reliability or battery lifetime limitations. As the power consumption depends heavily on the capacitances between adjacent wires, determining the optimal ordering and spacing of parallel wires is an important issue in the design of low power chips. As it turns out, optimal wire spacing is a convex optimization problem, whereas the optimal wire ordering is combinatorial in nature, containing (a special class of) the Minimum Hamilton Path problem. While the latter is \({\mathcal{NP}}\)-hard in general, the present paper provides an \({\mathcal{O}{(N \log N)}}\) algorithm that solves the coupled ordering and spacing problem for N parallel wires to optimality.

A DNA-based graph encoding scheme with its applications to graph isomorphism problems

Feynman first proposed DNA-based computation in 1961, but his idea was not implemented by experiment for a few decades. By properly manipulating DNA strands as the input instance of the Hamiltonian path problem, Adleman succeeded in solving the problem in a test tube. Since the experimental demonstration of its feasibility, DNA-based computing has been applied to a number of decision or combinatorial optimization problems. In this paper, we propose a DNA-based graph encoding scheme which can be used to solve some intractable graph problems, such as the subgraph isomorphism problem and its generalized problem – the maximum common subgraph problem, which are known to be NP-complete problems, in the Adleman–Lipton model using polynomial number of basic biological operations.

Maximally parallel attribute on P Systems: Properties and applications

In this paper, we investigate the maximally parallel attribute of P Systems. Some properties of P Systems are introduced, which are the filter property and the enumeration property. The two properties are applied to solving the sorting problem and the Hamilton cycle problem, respectively.

DNA media storage

In 1994, University of Southern California computer scientist, Dr. Leonard Adleman solved the Hamiltonian path problem using DNA as a computational mechanism. He proved the principle that DNA computing could be used to solve computationally complex problems. Because of the limitations in discovery time, resource requirements, and sequence mismatches, DNA computing has not yet become a commonly accepted practice. However, advancements are continually being discovered that are evolving the field of DNA computing. Practical applications of DNA are not restricted to computation alone. This research presents a novel approach in which DNA could be used as a means of storing files. Through the use of multiple sequence alignment combined with intelligent heuristics, the most probabilistic file contents can be determined with minimal errors.

Solutions to computational problems through gene assembly

Gene assembly in stichotrichous ciliates is an impressive computational process. They have a unique way of storing their genetic information in two fundamentally different forms within their two types of nuclei. Micronuclear genes are broken into blocks (called MDSs), with MDSs shuffled and separated by non-coding material; some of the MDSs may even be inverted. During gene assembly, all MDSs are sorted in the correct order to yield the transcription-able macronuclear gene. Based on the intramolecular model for gene assembly, we prove in this paper that gene assembly may be used in principle to solve computational problems. We prove that any given instance of the Hamiltonian path problem may be encoded in a suitable way in the form of an `artificial' gene so that gene assembly is successful on that gene-like pattern if and only if the given problem has an affirmative answer.

Combining Tree Partitioning, Precedence, and Incomparability Constraints

The tree constraint partitions a directed graph into node-disjoint trees. In many practical applications that involve such a partition, there exist side constraints specifying requirements on tree count, node degrees, or precedences and incomparabilities within node subsets. We present a generalisation of the tree constraint that incorporates such side constraints. The key point of our approach is to take partially into account the strong interactions between the tree partitioning problem and all the side constraints, in order to avoid thrashing during search. We describe filtering rules for this extended tree constraint and evaluate its effectiveness on three applications: the Hamiltonian path problem, the ordered disjoint paths problem, and the phylogenetic supertree problem.

Longest Path Problems on Ptolemaic Graphs

Longest path problem is a problem for finding a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, there are few known graph classes such that the longest path problem can be solved efficiently. Polynomial time algorithms for finding a longest cycle and a longest path in a Ptolemaic graph are proposed. Ptolemaic graphs are the graphs that satisfy the Ptolemy inequality, and they are the intersection of chordal graphs and distance-hereditary graphs. The algorithms use the dynamic programming technique on a laminar structure of cliques, which is a recent characterization of Ptolemaic graphs.

Minimum cycle factors in quasi-transitive digraphs

We consider the minimum cycle factor problem: given a digraph D , find the minimum number k min ( D ) of vertex disjoint cycles covering all vertices of D or verify that D has no cycle factor. There is an analogous problem for paths, known as the minimum path factor problem. Both problems are N P -hard for general digraphs as they include the Hamilton cycle and path problems, respectively. In 1994 Gutin [G. Gutin, Polynomial algorithms for finding paths and cycles in quasi-transitive digraphs, Australas. J. Combin. 10 (1994) 231–236] proved that the minimum path factor problem is solvable in polynomial time, for the class of quasi-transitive digraphs, and so is the Hamilton cycle problem. As the minimum cycle factor problem is analogous to the minimum path factor problem and is a generalization of the Hamilton cycle problem, it is therefore a natural question whether this problem is also polynomially solvable, for quasi-transitive digraphs. We conjecture that the problem of deciding, for a fixed k , whether a quasi-transitive digraph D has a cycle factor with at most k cycles is polynomial, and we verify this conjecture for k = 3 . We introduce the notion of an irreducible cycle factor and show how to convert a given cycle factor into an irreducible one in polynomial time when the input digraph is quasi-transitive. Finally, we show that even though this process will often reduce the number of cycles considerably, it does not always yield a minimum cycle factor.

Determinants and Longest Cycles of Graphs

We consider the Hamiltonian cycle problem on a given graph $G$. With such a graph we can associate a family $\mathcal{F}$ of probability transition matrices of Markov chains whose entries represent the probabilities of traversing corresponding arcs of the graph. When the underlying graph is Hamiltonian, we show the transition probability matrix induced by a Hamiltonian cycle maximizes—over $\mathcal{F}$—the determinant of a matrix that is a rank-one correction of the generator matrix of a Markov chain. In the case when the graph does not possess a Hamiltonian cycle, the above maximization yields a transition matrix of a chain with a longest simple cycle (in $G$) comprising that chain's unique ergodic class. These problems also have analogous eigenvalue interpretations.

Novel Ω-protocols for NP

Ω-protocols, introduced by Garay, Mackenzie and Yang, is a variant of S-protocols with online extractor which is a useful tool to overcome the nest effect in concurrent scenario. In this work, we construct an Ω-protocol for Hamiltonian cycle problem, and therefore, it allows us to present Ω-protocol for any NP relation. For most general NP relations, our construction of Ω-protocols is much more efficient than the informal one described by Garay et al. and we believe that the method for our construction may be of independent interest.

ON COMPUTING LONGEST PATHS IN SMALL GRAPH CLASSES

The longest path problem is the one that finds a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. Among those, for trees, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for some tree-like graph classes by this approach. We next propose two new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on these classes.

On finding spanning trees with few leaves

The problem of finding a spanning tree with few leaves is motivated by the design of communication networks, where the cost of the devices depends on their routing functionality (ending, forwarding, or routing a connection). Besides this application, the problem has its own theoretical importance as a generalization of the Hamiltonian path problem. Lu and Ravi showed that there is no constant factor approximation for minimizing the number of leaves of a spanning tree, unless P = NP . Thus instead of minimizing the number of leaves, we are going to deal with maximizing the number of non-leaves: we give a linear-time 2-approximation for arbitrary graphs, a 3 2 -approximation for claw-free graphs, and a 6 5 -approximation for cubic graphs.

Two New Classes of Hamiltonian Graphs

Abstract We prove that a triangular grid without local cuts is (almost) always Hamiltonian. This suggests an efficient scheme for rendering triangulated manifolds by graphics hardware. We also show that the Hamiltonian Cycle problem is NP-Complete for planar subcubic graphs of arbitrarily high girth. As a by-product we prove that there exist tri-Hamiltonian planar subcubic graphs of arbitrarily high girth.

On Hamilton Cycles in Locally Connected Graphs with Vertex Degree Constraints

Abstract It is shown that every connected, locally connected graph with the maximum vertex degree Δ ( G ) = 5 and the minimum vertex degree δ ( G ) ⩾ 3 is fully cycle extendable. For Δ ( G ) ⩽ 4 , all connected, locally connected graphs, including infinite ones, are explicitly described. The Hamilton Cycle problem for locally connected graphs with Δ ( G ) ⩽ 7 is shown to be NP-complete.

Solving the Hamiltonian path problem with a light-based computer

In this paper we propose a special computational device which uses light rays for solving the Hamiltonian path problem on a directed graph. The device has a graph-like representation and the light is traversing it by following the routes given by the connections between nodes. In each node the rays are uniquely marked so that they can be easily identified. At the destination node we will search only for particular rays that have passed only once through each node. We show that the proposed device can solve small and medium instances of the problem in reasonable time.