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Showing papers on "Longest path problem published in 2019"


Posted Content
TL;DR: It is seen immediately that the length of the longest path is also asymptotic to $f(c)n$ w.h.s.p.
Abstract: We discuss the length of the longest cycle in a sparse random graph $G_{n,p},p=c/n$. $c$ constant. We show that for large $c$ there is a function $f(c)$ such that $L_n(c)/n\to f(c)$ a.s. The function $f(c)=1-\sum_{k=1}^\infty p_k(c)e^{-kc}$ where $p_k$ is a polynomial in $k$. We are only able to explicitly give the values $p_1,p_2$, although we could in principle compute any $p_k$. We see immediately that the length of the longest path is also asymptotic to $f(c)n$ w.h.p.

11 citations


Journal ArticleDOI
TL;DR: This paper extends on existing models and extensively test new formulations for the graph-optimization problem, showing how one of the newly developed model clearly exhibits better performance, allowing to solve at optimality instances of significant sizes.

10 citations


Journal ArticleDOI
TL;DR: This paper considers an optimization problem for stochastic CPM problems, where task durations are expressed as discrete histograms obtained from historical operation data, that maximizes the probability that all tasks are completed within a given completion time by improving thetask durations on the critical path.
Abstract: The CPM (Critical Path Method) is a network-based approach for project management. This method identifies the longest path, which allows us to find the critical path that must be shortened so that the completion time of the whole project can be shortened. However, considering uncertainty in CPM is not straightforward. In this paper, we consider an optimization problem for stochastic CPM problems, where task durations are expressed as discrete histograms obtained from historical operation data, that maximizes the probability that all tasks are completed within a given completion time by improving the task durations on the critical path. We propose two reformulations of the problem as a mixed-integer linear programming problem: one based on tasks, and the other based on paths. In addition, we propose an iterative method to solve the problem efficiently by reducing the number of binary variables. Finally, we demonstrate efficiency of our proposed methods in some case studies.

9 citations


Journal ArticleDOI
TL;DR: In this article, the eigenvectors of Laplacian matrices of trees are reduced to a tridiagonal matrix using the Schur complement, and bounds on the ratio of eigenvector entries along a path are obtained.

9 citations


Journal ArticleDOI
TL;DR: A reverse greedy path is defined and it is shown both analytically and numerically that this scales with the logarithm of the size of the network with a coefficient given by the number of edges added using random attachment.
Abstract: The Price model, the directed version of the Barabasi-Albert model, produces a growing directed acyclic graph. We look at variants of the model in which directed edges are added to the new vertex in one of two ways: using cumulative advantage (preferential attachment) choosing vertices in proportion to their degree, or with random attachment in which vertices are chosen uniformly at random. In such networks, the longest path is well defined and in some cases is known to be a better approximation to geodesics than the shortest path. We define a reverse greedy path and show both analytically and numerically that this scales with the logarithm of the size of the network with a coefficient given by the number of edges added using random attachment. This is a lower bound on the length of the longest path to any given vertex and we show numerically that the longest path also scales with the logarithm of the size of the network but with a larger coefficient that has some weak dependence on the parameters of the model.

7 citations


Posted Content
TL;DR: 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.
Abstract: A supergrid graph is a finite vertex-induced subgraph of the infinite graph whose vertex set consists of all points of the plane with integer coordinates and in which two vertices are adjacent if the difference of their x or y coordinates is not larger than 1. The Hamiltonian path (cycle) problem is to determine whether a graph contains a simple path (cycle) in which each vertex of the graph appears exactly once. This problem is NP-complete for general graphs and it is also NP-complete for general supergrid graphs. Despite the many applications of the problem, it is still open for many classes, including solid supergrid graphs and supergrid graphs with some holes. A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In this paper, first we will study the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of rectangular supergrid graphs with a rectangular hole. Next, we will show that C-shaped supergrid graphs are Hamiltonian connected except few conditions. Finally, we will compute a longest path between two distinct vertices 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.

6 citations


Journal ArticleDOI
TL;DR: This paper extends the GHRV theorem to weighted mixed graphs and points out a new polynomial case where the edges form a clique, and yields a kernel and a parameterized algorithm that is slightly faster than the brute-force algorithm.
Abstract: Coloring of mixed graphs that contain both directed arcs and undirected edges is relevant for scheduling of unit-length jobs with precedence constraints and conflicts. The classic GHRV theorem (attributed to Gallai, Hasse, Roy, and Vitaver) relates graph coloring to longest paths. It can be extended to mixed graphs. In the present paper we further extend the GHRV theorem to weighted mixed graphs. As a byproduct this yields a kernel and a parameterized algorithm (with the number of undirected edges as parameter) that is slightly faster than the brute-force algorithm. The parameter is natural since the directed version is polynomial whereas the undirected version is NP-complete. Furthermore we point out a new polynomial case where the edges form a clique.

5 citations


Journal ArticleDOI
TL;DR: In this paper, a maximum length recalculation algorithm (MLRA) is proposed to update the length of the longest path to affected nodes in a perturbed DAG where multiple edges are simultaneously deleted and added.

5 citations


Proceedings Article
01 Jan 2019
TL;DR: An exact algorithm for solving the longest simple path problem between two given vertices in undirected weighted graphs is proposed by using graph partitioning and dynamic programming which yields the first efficient parallel algorithm for the problem.
Abstract: We propose an exact algorithm for solving the longest path problem between two given vertices in undirected weighted graphs. By using graph partitioning and dynamic programming, we obtain an algorithm that is significantly faster than other state-of-the-art methods. This enables us to solve instances that were previously unsolved and solve hard instances significantly faster. We also present a parallel version of the algorithm.

4 citations


Journal ArticleDOI
TL;DR: It is shown that a longest path between two given vertices s and t of an L-shaped grid graph can be computed in linear time.
Abstract: The longest path problem, that is, finding a simple path with the maximum number of vertices, is a well-known NP-hard problem with many applications. However, for some classes of graphs, including ...

4 citations


Journal ArticleDOI
TL;DR: This work shows that by estimating spatial orientation of the agents with single antenna, the accuracy is improved by 96% over crowdsourcing only, and solves the scheduling problem with a 3-approximation ratio in polynomial time.
Abstract: The distributed nature of policy violations in spectrum sharing necessitate the use of mobile autonomous agents (e.g., UAVs, self-driving cars, and crowdsourcing) to implement cost-effective enforcement systems. We define this problem as multiagent planning with cardinality (MPC), where cardinality represents multiple, unique agents visiting each infraction location to collectively improve the accuracy of the enforcement tasks. Designed as a practical and deployable system, our solution leverages crowdsourced information to determine the optimum cardinality and provide a routing schedule for the agents to achieve the desired level of accuracy of detection and localization at minimum possible cost. We show that by estimating spatial orientation of the agents with single antenna, the accuracy is improved by 96% over crowdsourcing only. Using geographical maps as the basis, we solve the scheduling problem with a 3-approximation ratio in polynomial time that exhibits statistically similar performance under variety of urban locale across multiple continents. The longest path traversed by an agent on average is 1.2 km per unit diagonal length of a rectangular geographic area, even when there are twice as many infractions as agents. Deploying UAVs to the estimated region of infraction improves localization accuracy by ≈70% compared to ground vehicles.

Journal ArticleDOI
TL;DR: In this paper, it was shown that for any probability distribution µ on N one can construct a stationary version of the infinite-bin model with move distribution µ, and that the growth rate C(p) of the longest path in a Barak-Erd˝ os graph of parameter p is analytic on (0, 1).
Abstract: In this article, we prove that for any probability distribution µ on N one can construct a stationary version of the infinite-bin model –an interacting particle system introduced by Foss and Konstantopoulos– with move distribution µ. Using this result, we obtain a new formula for the speed of the front of infinite-bin models, as a series of positive terms. This implies that the growth rate C(p) of the longest path in a Barak-Erd˝ os graph of parameter p is analytic on (0, 1].

Posted Content
TL;DR: A natural hierarchy of space complexity classes of FPS, SubPS, SemiPS, SupPS and BrutePS is defined, and tight classifications for several well-studied graph problems such as Longest Path, Feedback Vertex Set, Dominating Set, Girth, Treewidth, etc. into this hierarchy are obtained.
Abstract: Parameterized complexity attempts to give a more fine-grained analysis of the complexity of problems: instead of measuring the running time as a function of only the input size, we analyze the running time with respect to additional parameters. This approach has proven to be highly successful in delineating our understanding of \NP-hard problems. Given this success with the TIME resource, it seems but natural to use this approach for dealing with the SPACE resource. First attempts in this direction have considered a few individual problems, with some success: Fafianie and Kratsch [MFCS'14] and Chitnis et al. [SODA'15] introduced the notions of streaming kernels and parameterized streaming algorithms respectively. For example, the latter shows how to refine the $\Omega(n^2)$ bit lower bound for finding a minimum Vertex Cover (VC) in the streaming setting by designing an algorithm for the parameterized $k$-VC problem which uses $O(k^{2}\log n)$ bits. In this paper, we initiate a systematic study of graph problems from the paradigm of parameterized streaming algorithms. We first define a natural hierarchy of space complexity classes of FPS, SubPS, SemiPS, SupPS and BrutePS, and then obtain tight classifications for several well-studied graph problems such as Longest Path, Feedback Vertex Set, Dominating Set, Girth, Treewidth, etc. into this hierarchy. (see paper for full abstract)

Proceedings ArticleDOI
01 Sep 2019
TL;DR: A planner that generates paths for multiple unmanned aerial vehicles (UAVs) that need to cover multiple disjoint non-convex polygonal areas with nadir pointing cameras of varying field of view is proposed.
Abstract: Rescue and security applications call for fast mapping of complex environments. For this purpose we propose a planner that generates paths for multiple unmanned aerial vehicles (UAVs) that need to cover multiple disjoint non-convex polygonal areas with nadir pointing cameras of varying field of view. These fulfill a user specified overlap to allow for taken images being used in different map generation procedures. The proposed planner first decomposes the given areas into suitably sized convex subareas. These are allocated to the available UAVs by a metaheuristic solver while minimizing the longest path of all involved UAVs. Given the allocation, subareas are adjusted by a developed heuristic to satisfy the overlap constraints at the inner borders of the split areas. Finally paths allowing for scanning the areas are generated. Experiments are conducted to demonstrate and validate the results of the planner.

Proceedings ArticleDOI
01 Oct 2019
TL;DR: A linear-time algorithm is proposed to find the longest (s, t)-path of O-shaped supergrid graphs, which can be used to compute the smallest stitching path of computerized embroidery machine and 3D printer when a hollow object is printed.
Abstract: The longest path and Hamiltonian problems were known to be NP-complete. In spite of many applications of these problems, their complexities are still unknown for many classes of graphs, including supergrid graphs with holes and solid supergrid graphs. In this paper, we will study the complexity of the longest (s, t)-path problem on O-shaped supergrid graphs. The longest (s, t)-path is a simple path from s to t with the largest number of visited vertices. An O-shaped supergrid graph is a rectangular supergrid graph with one rectangular hollow. We will propose a linear-time algorithm to find the longest (s, t)-path of O-shaped supergrid graphs. The longest (s, t)-paths of O-shaped supergrid graphs can be used to compute the smallest stitching path of computerized embroidery machine and 3D printer when a hollow object is printed.

Journal ArticleDOI
26 Jun 2019
TL;DR: In this paper, the authors presented a graph embeddable into Cubic lattices L3, such that graphs can also occur as sub graphs of the cubic lattices, and enjoying the property that every vertex is missed by some longest path.
Abstract: Tibor Gallai in 1966 elevated the declaration about the existence of graphs with the property that every vertex is missed by some longest path. This property will be called Gallai’s property. First answer back by H. Walther, who introduced a planar graph on 25 vertices satisfying Gallai’s property, and various authors worked on that property, after examples of such graphs were found while examining such n-dimensional Ln graphs with the property that every longest Paths have empty intersection, can be embeddable in IRn, Some in equilateral triangular lattice T, Square lattice L2, hexagonal lattice H, also on the torus, Mobius strip, and the Klein bottle but no hypo-Hamiltonian graphs are embeddable in the planar square lattice. In this paper we present a graph embeddable into Cubic lattices L3, such that graphs can also occur as sub graphs of the cubic lattices, and enjoying the property that every vertex is missed by some longest path. Here research has also significance in applications. What if several processing units are interlinked as parts of a lattice network. Some of them developing a chain of maximal length are used to solve a certain task. To get a self-stable fault-tolerant system, it is indispensable that in case of failure of any unit or link, another chain of same length, not containing the faulty unit or link, can exchange the chain originally used. This is exactly the case investigated here. We denote by Ln the n-dimensional cubic lattice in IRn.

Book ChapterDOI
27 May 2019
TL;DR: A way to incorporate the similarity of states into minLP using different distance functions to improve dual bounds for the maximum independent set problem (MISP) and the set cover problem (SCP), providing empirical evidence for this assumption.
Abstract: Over the last years, binary decision diagrams (BDDs) have become a powerful tool in the field of combinatorial optimization. They are directed acyclic multigraphs and represent the solution space of binary optimization problems in a recursive way. During their construction, merging of nodes in this multigraph is applied to keep the size within polynomial bounds resulting in a discrete relaxation of the original problem. The longest path length through this diagram corresponds then to an upper bound of the optimal objective value. The algorithm deciding which nodes to merge is called a merging heuristic. A commonly used heuristic for layer-wise construction is minimum longest path length (minLP) which sorts the nodes in a layer descending by the currently longest path length to them and subsequently merges the worst ranked nodes to reduce the width of a layer. A shortcoming of this approach is that it neglects the (dis-)similarity between states it merges, which we assume to have negative impact on the quality of the finally obtained bound. By means of a simple tie breaking procedure, we show a way to incorporate the similarity of states into minLP using different distance functions to improve dual bounds for the maximum independent set problem (MISP) and the set cover problem (SCP), providing empirical evidence for our assumption. Furthermore, we extend this procedure by applying similarity-based node merging also to nodes with close but not necessarily identical longest path values. This turns out to be beneficial for weighted problems where ties are substantially less likely to occur. We evaluate the method on the weighted MISP and tune parameters that control as to when to apply similarity-based node merging.

Journal ArticleDOI
TL;DR: In this paper, a structural characterization of minimal vertex separators in cographs is presented and polynomial-time algorithms for some connectivity augmentation problems and its variants are proposed.
Abstract: The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs These graphs can be represented in a tree-like representation termed parse tree or cotree In this paper, we study some popular combinatorial problems restricted to cographs We first present a structural characterization of minimal vertex separators in cographs Further, we show that listing all minimal vertex separators and finding some constrained vertex separators are linear-time solvable in cographs We propose polynomial-time algorithms for some connectivity augmentation problems and its variants in cographs, preserving the cograph property Finally, using the dynamic programming paradigm, we present a generic framework to solve classical optimization problems such as the longest path, the Steiner path and the minimum leaf spanning tree problems restricted to cographs and our framework yields polynomial-time algorithms for the three problems

Patent
04 Jun 2019
TL;DR: In this article, the authors present a technique for job flow processing based on computation of independent and dependent metrics for tasks in a job flow, such as time remaining until end and longest path to end.
Abstract: Various embodiments are generally directed to techniques for job flow processing, such as by ordering the performance of parallel tasks in a job flow to minimize a makespan for the job flow, for instance. Some embodiments are particularly directed to ordering the performance of tasks in a job flow based on computation of one or more independent and dependent metrics for tasks in a job flow. In many embodiments, tasks along a critical path of a job flow may be identified and prioritized using the one or more metrics computed for tasks in the job flow. For example, computing a time remaining until end and/or a longest path to end for each task in a job flow may enable a listing of tasks in the job flow to be ordered in a manner that prioritizes tasks to optimize the makespan for the job flow to be executed.

Posted Content
TL;DR: In undirected graphs, it is shown that MBT has no efficient $\exp(-O(\log^{0.63}{n}))$-approximation under the exponential time hypothesis, and inapproximability results rely on self-improving reductions and structural properties of binary trees.
Abstract: We introduce and investigate the approximability of the maximum binary tree problem (MBT) in directed and undirected graphs. The goal in MBT is to find a maximum-sized binary tree in a given graph. MBT is a natural variant of the well-studied longest path problem, since both can be viewed as finding a maximum-sized tree of bounded degree in a given graph. The connection to longest path motivates the study of MBT in directed acyclic graphs (DAGs), since the longest path problem is solvable efficiently in DAGs. In contrast, we show that MBT in DAGs is in fact hard: it has no efficient $\exp(-O(\log n/ \log \log n))$-approximation algorithm under the exponential time hypothesis, where $n$ is the number of vertices in the input graph. In undirected graphs, we show that MBT has no efficient $\exp(-O(\log^{0.63}{n}))$-approximation under the exponential time hypothesis. Our inapproximability results rely on self-improving reductions and structural properties of binary trees. We also show constant-factor inapproximability assuming $\text{P} eq \text{NP}$. In addition to inapproximability results, we present algorithmic results along two different flavors: (1) We design a randomized algorithm to verify if a given directed graph on $n$ vertices contains a binary tree of size $k$ in $2^k \text{poly}(n)$ time. (2) Motivated by the longest heapable subsequence problem, introduced by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011), which is equivalent to MBT in permutation DAGs, we design efficient algorithms for MBT in bipartite permutation graphs.

Proceedings ArticleDOI
26 Nov 2019
TL;DR: In this paper, it was shown that the problem is fixed-parameter tractable (FPT) on undirected graphs and actually admits a single-exponential algorithm, that is, one of running time exp(O(k)) * poly(n).
Abstract: We consider the following natural "above guarantee" parameterization of the classical longest path problem: For given vertices s and t of a graph G, and an integer k, the longest detour problem asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that the longest detour problem is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) * poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k. Furthermore, we study a related problem, exact detour, that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k * poly(n), and a deterministic algorithm with running time about 6.745^k * poly(n), showing that this problem is FPT as well. Our algorithms for the exact detour problem apply to both undirected and directed graphs.

Journal ArticleDOI
17 Sep 2019
TL;DR: In this article, a graph with the property that each vertex is missed by some longest cycle with connectivity 2 is presented, which is called Gallai's property and can be used to construct Hypo-traceable graphs.
Abstract: The most famous examples of Hypo-Hamiltonian graph is the Petersen graph. Before the discovery of Hypo-traceable graphs, Tibor Gallai, in 1966, raised the question whether the graphs in which each vertex is missed by some longest path. This property will be called Gallai’s property, various authors worked on that property. In 1969, Gallai’s question was first replied through H. Walther, who introduced a planar graph on 25 vertices satisfying Gallai’s criterion. Furthermore, H. Walther and H. Voss and Tudor Zamfirescu introduced the graph with 12 vertices and it was guessed that order 12 is the smaller possibility of such a graphLater the question was modifies by Tudor Zamfirescu and asked that whether there exists graphs of Paths and Cycles, that is to say i-connected graphs (planar or non-planar respectively), such that each set of j points are disjoint from some longest paths or cycles., Several good examples answering Tudor Zamfirescu’s questions were published. In this note a graphs is developed with the property that everyone vertex is missed by some longest cycle with connectivity 2, satisfying Gallai’s property. The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.

Book ChapterDOI
10 Sep 2019
TL;DR: This work proposes a prediction mechanism to evaluate a set of different merging mechanisms at each layer during the construction of a relaxed BDD, in order to always select and apply the most promising heuristic.
Abstract: Relaxed binary decision diagrams (BDDs) are used in combinatorial optimization as a compact representation of a relaxed solution space. They are directed acyclic multigraphs which are derived from the state space of a recursive dynamic programming formulation of the considered optimization problem. The compactness of a relaxed BDD is achieved by superimposing states, which corresponds to merging BDD nodes in the classical layer-wise top-down BDD construction. Selecting which nodes to merge crucially determines the quality of the resulting BDD and is the task of a merging heuristic, for which the minimum longest path value (minLP) heuristic has turned out to be highly effective for a number of problems. This heuristic sorts the nodes in a layer by decreasing current longest path value and merges the necessary number of worst ranked nodes into one. There are, however, also other merging heuristics available and usually it is not easy to decide which one is more promising to use in which situation. In this work we propose a prediction mechanism to evaluate a set of different merging mechanisms at each layer during the construction of a relaxed BDD, in order to always select and apply the most promising heuristic. This prediction is implemented by either a perfect or by a k-layers lookahead construction of the BDD, gathering feature vectors for two competing merging heuristics which are then fed into a binary classifier. Models based on statistical tests and a feed-forward neural network are considered for the classifier. We study this approach for the maximum weighted independent set problem and in conjunction with a parameterized merging heuristic that takes also the similarity between states into account. We train and validate the binary classifiers on random graphs and finally test on weighted DIMACS instances. Results indicate that relaxed BDDs can be obtained whose upper bounds are on average up to \(\approx \)16% better than those of BDDs constructed with the sole use of minLP.

Proceedings ArticleDOI
01 Aug 2019
TL;DR: Emulation results show the counterintuitive result that the size of the maze has a negligible influence on the time the solution is found, as well as exploiting the wave digital concept to create a real-time capable algorithm that has a direct correspondence to an electrical circuit.
Abstract: Since electrical circuits are known to be inherently massively parallel architectures, circuit-inspired approaches are potential candidates to solve computationally complex problems. One of these problems is discovering all possible paths through a maze, which includes the shortest and the longest path, where finding the latter is known to be a np-complete problem. The key towards an efficient solution to the problem are self-organizing memristive circuits. Utilizing memristors – resistors with a memory – has shown to outperform all existing graph-theoretical algorithms and while theoretical models of memristors have been used to tackle the maze problem before, it has not been addressed by using models of real, manufacturable memristors. This paper intends to employ physical models of a real resistive random access memory (RRAM) memristive devices that are known for their fast switching behavior to find the longest path through a maze. We exploit the wave digital concept to create a real-time capable algorithm that has a direct correspondence to an electrical circuit. The emulation results show the counterintuitive result that the size of the maze has a negligible influence on the time the solution is found.

Journal ArticleDOI
TL;DR: A genetic algorithm using a modification of the edge recombination operator is presented to find the longest snake in dimensions one to nine with lesser edge failures.

Journal ArticleDOI
25 Dec 2019
TL;DR: In this article, a more general sharp bound for the circumference of a longest path in a 2-connected graph was presented, including the bound c ǫ as an immediate corollary, based on elementary arguments.
Abstract: Let l be the length of a longest path in a 2-connected graph G and c the circumference - the length of a longest cycle in G In 1952, Dirac proved that c , by noting that "actually c , but the proof of this result, which is best possible, is rather complicated" Let L1;L2; :::;Lm be a vine on a longest path of G In this paper, using the parameter m, we present a more general sharp bound for the circumference c including the bound c as an immediate corollary, based on elementary arguments

Journal ArticleDOI
TL;DR: It is shown that a simple algorithm, based on depth-first search, finds on almost every undirected graph a path of length more than 3-3\sqrt{|V| \log |V|} and so has performance ratio less than 1.
Abstract: The longest path problem is known to be NP-hard. Moreover, they cannot be approximated within a constant ratio, unless ${\\rm P=NP}$. The best known polynomial time approximation algorithms for this problem essentially find a path of length that is the logarithm of the optimum.In this paper we investigate the performance of an approximation algorithm for this problem in almost every case. We show that a simple algorithm, based on depth-first search, finds on almost every undirected graph $G=(V,E)$ a path of length more than $|V|-3\\sqrt{|V| \\log |V|}$ and so has performance ratio less than $1+4\\sqrt{\\log |V|/|V|}$.\\

Patent
13 Dec 2019
TL;DR: In this article, a cloud manufacturing service combination optimization method based on a k-Dijkstra algorithm is proposed, and the optimization of the combined service under the cloud manufacturing mode is realized, and by optimizing the service time, the service cost, the manufacturing capability and the comprehensive capability, the Cloud manufacturing enterprise can effectively helped to reduce the production cost and improve the production efficiency.
Abstract: The invention provides a cloud manufacturing service combination optimization method based on a k _ Dijkstra algorithm. According to the method, a cloud manufacturing service combination optimizationproblem is converted into a standard directed graph problem; for a standard directed graph, according to a sub-short path theorem, a path extension method and a Dijkstra algorithm, first k shortest path algorithms under a cloud manufacturing mode are provided, namely the k _ Dijkstra algorithm; an algorithm of the first k longest paths is provided; the first k shortest path algorithms and the first k longest path algorithms under the cloud manufacturing mode are provided by combining the first k shortest path algorithms and the first k longest path algorithms. The optimization of the combinedservice under the cloud manufacturing mode is realized, and by optimizing the service time, the service cost, the manufacturing capability and the comprehensive capability, the cloud manufacturing enterprise can be effectively helped to reduce the production cost and improve the production efficiency.

Journal ArticleDOI
TL;DR: In this paper, an algorithm for optimal hamiltonian coloring of a special class of block graphs, namely $SDB(p/2)$ block graphs is presented and characterized level-wise regular block graphs and extended star of blocks achieving this lower bound.
Abstract: Let $G$ be a simple connected graph of order $n$. A hamiltonian coloring $c$ of a graph $G$ is an assignment of colors (non-negative integers) to the vertices of $G$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n - 1$ for every two distinct vertices $u$ and $v$ of $G$, where $D(u, v)$ denotes the detour distance between $u$ and $v$ in $G$ which is the length of the longest path between $u$ and $v$. The value \emph{hc(c)} of a hamiltonian coloring $c$ is the maximum color assigned to a vertex of $G$. The hamiltonian chromatic number, denoted by $hc(G)$, is min\{$hc(c)$\} taken over all hamiltonian coloring $c$ of $G$. In this paper, we give a necessary and sufficient condition to achieve a lower bound for the hamiltonian chromatic number of block graphs given in [Theorem 1,On Hamiltonian Colorings of Block graphs, In: Kaykobad, M., Petrechi, R., (eds.) WALCOM: Algorithms and Computation, LNCS: 9627, 28-39, 2016]. We present an algorithm for optimal hamiltonian coloring of a special class of block graphs, namely $SDB(p/2)$ block graphs. We characterize level-wise regular block graphs and extended star of blocks achieving this lower bound.

Journal ArticleDOI
TL;DR: This paper presents a necessary and sufficient condition for the existence of a Hamiltonian cycle in convex bipartite graphs, and further, a linear-time algorithm is obtained for this graph class.
Abstract: A convex bipartite graph G with bipartition (X, Y) and an ordering $$X=(x_1,\ldots ,x_n)$$ is a bipartite graph such that for each $$y \in Y$$ , the neighborhood of y in X appears consecutively G is said to have convexity with respect to X In this paper, we present a necessary and sufficient condition for the existence of a Hamiltonian cycle in convex bipartite graphs, and further, we obtain a linear-time algorithm for this graph class We also show that Chvatal’s necessary condition is sufficient for convex bipartite graphs The closely related problem is HAMILTONIAN PATH whose complexity is open in convex bipartite graphs We classify the class of convex bipartite graphs as monotone and non-monotone graphs For monotone convex bipartite graphs, we present a linear-time algorithm to output a Hamiltonian path It is important to highlight that: (a) in Keil (Inf Process Lett 20:201–206, 1985) and Ghosh (in: WALCOM 2011, LNCS 6552, pp 191–201, 2011), it is incorrectly claimed that Hamiltonian path problem in convex bipartite graphs is polynomial-time solvable by referring to Muller (Discrete Math 156:291–298, 1996) which actually discusses Hamiltonian cycle and (b) the algorithm appeared in Ghosh (2011) for the longest path problem (Hamiltonian path problem) in biconvex and convex bipartite graphs has an error and it does not compute an optimum solution always We present an infinite set of counterexamples in support of our claim