The One-Fault Dimension-Balanced Hamiltonian Problem in Toroidal Mesh Graphs

Hamiltonian cycle problems play a central role in graph theory and have wide-ranging applications in network-on-chip architectures, interconnection networks, and large-scale parallel systems. When a network contains faulty nodes or faulty links, the feasibility of certain paths becomes restricted, making the construction of Hamiltonian cycles substantially more difficult and increasingly important for ensuring reliable communication. A dimension-balanced Hamiltonian cycle is a special type of cycle that maintains an even distribution of edges across multiple dimensions of a network. Its directed counterpart extends this idea to symmetric directed networks by balancing the number of edges used in each positive and negative direction. Such cycles are desirable because they support uniform traffic distribution and reduce communication contention in practical systems. Previous research has examined the existence of directed dimension-balanced Hamiltonian cycles in directed toroidal mesh networks and has shown that some configurations permit directed dimension-balanced Hamiltonian cycles while others do not. Building on this foundation, this paper investigates the fault-tolerant properties of such networks by analyzing whether directed dimension-balanced Hamiltonian cycles still exist when a single vertex (node) or a single edge (link) is faulty. Our results extend the current understanding of Hamiltonian robustness in symmetric directed networks.

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

A digraph is semicomplete if it has no pair of non-adjacent vertices. It is complete if every pair of distinct vertices induces a 2-cycle. A digraph is semicomplete multipartite if it can be obtained from a semicomplete digraph D by choosing a collection of vertex-disjoint subsets X₁,…,X_c of V(D) and then deleting all arcs both of whose end-vertices lie inside some X_i. We can also think of a semicomplete digraph as being obtained from a semicomplete multipartite digraph on the same vertex set and partite sets V₁,…,V_c by adding the arcs of a semicomplete digraph D_i on V_i for each partite set V_i. It is well known that both the hamiltonian path and the hamiltonian cycle problem can be solved in polynomial time for semicomplete multipartite digraphs. In this paper we study the complexity of finding a hamiltonian path or cycle in a semicomplete digraph S which is obtained as above from a semicomplete multipartite digraph D and semicomplete digraphs D_i=(V_i,A_i), i∈[c] such that the path or cycle uses as few arcs of A₁∪…A_c as possible. We obtain a number of results for the case when each D_i is a complete digraph. Already this case is highly nontrivial in the cycle case and the complexity is still open. We show how to find a Hamiltonian path which uses as few arcs from the D_i’s as possible in polynomial time and obtain a number of results, both structural and algorithmic on hamiltonian cycles that use the minimum or close to the minimum number of arcs from the D_i’s. Our results imply the polynomial solvability of some special cases of the NP-complete {0,1}-TSP problem. Finally we show that two natural questions about properties of quasi-hamiltonian cycles, that is, cycles meeting all partite sets in semicomplete multipartite digraphs are NP-complete.

Kombinatorische entscheidungsprobleme: Methoden und anwendungen; fortbildungskurs des instituts für operations research der ETH Zürich: Thomas M. Liebling, Max Rössler (Eds.) Springer, Berlin, 1978, 206 pages, Dfl. 32.75 Soft cover

Molecular computing has proved its possibility to solve weighted graph problem such as Hamiltonian Path Problem (HPP), Traveling Salesman Problem (TSP) and the Shortest Path Problem (SPP). Normally, in molecular computation, the DNA sequences used for the computation should be critically designed in order to reduce error that could occur during computation. In the previous paper, we have proposed a readout method tailored specifically to HPP in DNA Computing using real-time PCR for output visualization. Six nodes of HPP was considered. Based on the example instance, the method requires 11 oligonucleotides, where 6 oligonucleotides are for the nodes and 5 oligonucleotides are for the edges. Three TaqMan probes and five primers are required as well. In this study, a procedure for DNA sequence design is presented in order to obtain good sequences for those nodes, primers, and probes. The experiment is done based on the generated DNA sequences and the Hamiltonian Path can be determined successfully.

We consider the complexity of the Hamilton cycle decision problem when restricted to k-uniform hypergraphs H of high minimum codegree delta(H). We show that for tight Hamilton cycles this problem is NP-hard even when restricted to k-uniform hypergraphs H with delta(H) >= n/2 - C, where n is the order of H and C is a constant which depends only on k. This answers a question raised by Karpinski, Rucinski and Szymanska. Additionally we give a polynomial-time algorithm which, for a sufficiently small constant epsilon > 0, determines whether or not a 4-uniform hypergraph H on n vertices with delta(H) >= n/2 - epsilon * n contains a Hamilton 2-cycle. This demonstrates that some looser Hamilton cycles exhibit interestingly different behaviour compared to tight Hamilton cycles. A key part of the proof is a precise characterisation of all 4-uniform hypergraphs H on n vertices with delta(H) >= n/2 - epsilon * n which do not contain a Hamilton 2-cycle; this may be of independent interest. As an additional corollary of this characterisation, we obtain an exact Dirac-type bound for the existence of a Hamilton 2-cycle in a large 4-uniform hypergraph.

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.

Hamiltonian problems in directed graphs with simple row patterns

The crossed cube is a prominent variant of the well known, highly regular-structured hypercube. In [24], it is shown that due to the loss of regularity in link topology, generating Hamiltonian cycles, even in a healthy crossed cube, is a more complicated procedure than in the hypercube, and fewer Hamiltonian cycles can be generated in the crossed cube. Because of the importance of fault-tolerance in interconnection networks, in this paper, we treat the problem of embedding Hamiltonian cycles into a crossed cube with failed links. We establish a relationship between the faulty link distribution and the crossed cube's tolerability. A succinct algorithm is proposed to find a Hamiltonian cycle in a CQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> tolerating up to n-2 failed links.

This dissertation is to illustrate the current state of the art of DNA computing achievements, especially of new approaches to solve theoretical 3-SAT problems and the hitting-set problem. Beginning with Adleman’s breakthrough which is an molecular algorithm for the solution of a NP-complete, combinatorial problem, the directed Hamiltonian path problem (HPP). Today, many researchers all over the world concentrate on proposing new methods to solve engineering or application problems with a DNA computing approach. Satisfiability problem is given a Boolean formula, and decide if a satisfying truth assignment exists. ( ) ( ) … ( ) ( ) is an example of Boolean formula. k-SAT problem means that each clause has exactly k literals. Not-All-Equal (NAE) 3-SAT problem and One-In-Three (1IN3) 3-SAT problem are both NP-complete problems. In this dissertation, we present molecular solutions to find all true assignments (3-SAT problem) and furthermore find Not-All-Equal (NAE) solutions and One-In-Three (1IN3) solutions in DNA-based Supercomputing. Hitting-set problem assume that there exists a collection C of subsets of a finite set S, and a positive integer K  |S|, and we need to know if there is a subset  S with | |  K such that contains at least one element of each subset in C. In other words, is the subset that intersects every subset in C and is called the hitting-set. In this dissertation, a DNA-based algorithm is proposed to solve the small hitting-set problem. A small hitting-set is a hitting-set with the smallest K value, i.e., the hitting-set with the smallest number of elements. Furthermore, an algorithm is introduced to find the number of ones from 2n combinations and minimum numbers of ones represents the small hitting-set since K is expected to be as small as possible. The complexity of all the presented DNA-based algorithms is also discussed. We describe time complexity and volume complexity of three Algorithms, numbers of test tube used and the longest library strand in solution space of all three Algorithms. Finally, the simulated experiment is applied to verify correctness of the proposed DNA-based algorithm for solving the One-In-Three (1IN3) 3-SAT problem, and simulation of Not-All-Equal (NAE) 3-SAT problem is similar. Also, another simulated experiment is applied to our proposed DNA-based algorithm 6-2, in order to solve the well-known hitting-set problem. This research has been motivated by the benefit and the application of DNA computing and gives new methods to solve two 3-SAT problems and the hitting-set problem which are NP-complete.

We 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

A number of combinatorial problems are treated using properties of abelian null-square-generated and idempotent-generated subalgebras of Clifford algebras. For example, the problem of deciding whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of Λk, where Λ is an appropriate nilpotent adjacency matrix, the k-cycles in any finite graph are recovered. Within the algebra context (i.e., considering the number of multiplications performed within the algebra), these problems are reduced to matrix multiplication, which is in complexity class P. The Hamiltonian cycle problem is one of many problems moved from classes NP-complete and #P-complete to class P in this context. Other problems considered include the set covering problem, counting the edge-disjoint cycle decompositions of a finite graph, computing the permanent of an arbitrary matrix, computing the girth and circumference of a graph, and finding the longest path in a graph.

The Hamiltonian cycle and path problems are fundamental in graph theory and useful in modelling real-life problems. Research in this area is directed toward designing better and better algorithms for general problems, but also toward defining new special cases for which exact polynomial-time algorithms exist. In the paper, such new classes of digraphs are proposed. The classes include, among others, quasi-adjoint graphs, which are a superclass of adjoints, directed line graphs, and graphs modelling a DNA sequencing problem.

We study the Hamilton cycle problem with input a random graph G=G(n,p) in two settings. In the first one, G is given to us in the form of randomly ordered adjacency lists while in the second one we are given the adjacency matrix of G. In each of the settings we give a deterministic algorithm that w.h.p. either it finds a Hamilton cycle or it returns a certificate that such a cycle does not exists, for p > 0. The running times of our algorithms are w.h.p. O(n) and O(n/p) respectively each being best possible in its own setting.

The dominating graph of a graph H has as its vertices all dominating sets of H, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of H. In this paper we prove that the dominating graph of any tree has a Hamilton path. We also show how a result about binary strings leads to a proof that the dominating graph of a cycle on n vertices has a Hamilton path if and only if n is not a multiple of 4.