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Showing papers on "Longest path problem published in 2018"


Journal ArticleDOI
TL;DR: In this article, the controllability of multiagent systems based on path and cycle graphs was studied and sufficient and necessary conditions were presented for determining the locations of leaders under which the controLLability can be realized.
Abstract: Summary This paper studies the controllability of multiagent systems based on path and cycle graphs. For paths or cycles, sufficient and necessary conditions are presented for determining the locations of leaders under which the controllability can be realized. Specifically, the controllability of a path is shown to be determined by a set generated only from its odd factors, and the controllability of a cycle is determined by whether the distance between 2 leaders belongs to a set generated from its even (odd) factors when the number of its nodes is even (odd). For both graphs, the dimension of the controllable subspace is also derived. Moreover, the technique used in the derivation of the above results is further used to get sufficient and necessary conditions for several different types of graphs generated from path and cycle topologies. These types of graphs can be regarded as typical topologies in the study of multiagent controllability, and accordingly the obtained results have meaningful enlightenment for the research in this field.

70 citations


Journal ArticleDOI
TL;DR: A concept of robust mean-excess travel time is introduced to hedge against the risk from both the uncertainty of the random travel times and the uncertainty in their distributions and the impact of uncertainty on the relative benefit and cost of robust paths is demonstrated.
Abstract: Routing service considering uncertainty is at the core of intelligent transportation systems and has attracted increasing attention. Existing stochastic shortest path models require the exact probability distributions of travel times and usually assume that they are independent. However, the distributions are often unavailable or inaccurate due to insufficient data, and correlation of travel times over different links has been observed. This paper presents a robust shortest path (RSP) model that only requires partial distribution information of travel times, including the support set, mean, variance, and correlation matrix. We introduce a concept of robust mean-excess travel time to hedge against the risk from both the uncertainty of the random travel times and the uncertainty in their distributions. To solve the RSP problem, an equivalent dual formulation is derived and used to design tight lower and upper bound approximation methods, which adopt the scenario approach and semi-definite programming approach, respectively. To solve large problems, we further propose an efficient primal approximation method, which only needs to solve two deterministic shortest path problems and a mean-standard deviation shortest path problem, and analyze its approximation performance. Experiments validate the tightness of the proposed bounds and demonstrate the impact of uncertainty on the relative benefit and cost of robust paths.

49 citations


Journal ArticleDOI
TL;DR: A comprehensive analysis of the influence of the path on the simulation errors is presented, based on which guidelines for choosing an optimal path were developed, and indicates that the optimal path is defined as the one minimizing the information lost by the omission of neighbors.
Abstract: Sequential Gaussian Simulation is a commonly used geostatistical method for populating a grid with a Gaussian random field. The theoretical foundation of this method implies that all previously simulated nodes, referred to as neighbors, should be included in the kriging system of each newly simulated node. This would, however, require solving a large number of linear systems of increasing size as the simulation progresses, which, for computational reasons, is generally not feasible. Traditionally, this problem is addressed by limiting the number of neighbors to the ones closest to the simulated node. This does, however, result in artifacts in the realization. The simulation path, that is, the order in which nodes are visited, is known to influence the location and magnitude of these artifacts. So far, few rigorous studies linking the simulation path to the associated biases are available and, correspondingly, recommendations regarding the choice of the simulation path are largely based on empirical evidence. In this study, a comprehensive analysis of the influence of the path on the simulation errors is presented, based on which guidelines for choosing an optimal path were developed. The most common path types are systematically assessed based on the comparison of the simulation covariance matrices with the covariance of the underlying spatial model. Our analysis indicates that the optimal path is defined as the one minimizing the information lost by the omission of neighbors. Classification into clustering paths, that is, paths simulating consecutively close nodes, and declustering paths, that is, paths simulating consecutively distant nodes, was found to be an efficient way of determining path performance. Common examples of the latter are multi-grid, mid-point, and quasi-random paths, while the former include row-by-row and spiral paths. Indeed, clustering paths tend to inadequately approximate covariances at intermediate and large lag distances, because their neighborhood is only composed of nearby nodes. On the other hand, declustering paths minimize the correlation among nodes, thus ensuring that the neighbors are more diverse, and that only weakly correlated neighbors are omitted.

30 citations


Proceedings ArticleDOI
20 Jun 2018
TL;DR: An algorithm is devised that approximately computes the number of paths of length k in a given directed graph with n vertices up to a multiplicative error of 1 ± ε, and a deterministic 2k·poly(n) time algorithm to find a k-path in aGiven directed graph that is promised to have few of them is obtained.
Abstract: We devise an algorithm that approximately computes the number of paths of length k in a given directed graph with n vertices up to a multiplicative error of 1 ± e. Our algorithm runs in time e−2 4k(n+m) poly(k). The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic 2k·poly(n) time algorithm to find a k-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect k-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.

19 citations


Book ChapterDOI
01 Jan 2018
TL;DR: A novel Genetic Algorithm-based solution to the critical path problem in Software Project Management, which is NP as against the shortest path problem, and has been implemented and verified using benchmarks.
Abstract: The critical path problem, in Software Project Management, finds the longest path in a Directed Acyclic Graph. The problem is immensely important for scheduling the critical activities. The problem reduces to the longest path problem, which is NP as against the shortest path problem. The longest path is an important NP-hard problem, which finds its applications in many other areas like graph drawing, sequence alignment algorithms, etc. The problem has been dealt with using Computational Intelligence. The paper presents the state of the art. The applicability of Genetic Algorithms in longest path problem has also been discussed. This paper proposes a novel Genetic Algorithm-based solution to the problem. This algorithm has been implemented and verified using benchmarks. The results are encouraging.

13 citations


Journal ArticleDOI
TL;DR: This article intends to solve a proposed multi-criteria shortest path problem of a weighted connected directed network whose associated edge weights are represented as rough variables in order to tackle the imprecision.
Abstract: Shortest path problem in real life applications has to deal with multiple criteria. This article intends to solve a proposed multi-criteria shortest path problem of a weighted connected directed network whose associated edge weights are represented as rough variables in order to tackle the imprecision. We have exhibited two different approaches to determine the optimum path(s) of the proposed problem. The first approach is the proposed modified rough Dijkstra’s labelling algorithm. The second approach considers the rough chance constrained programming technique to formulate the proposed multi-criteria shortest path problem which is eventually solved by two different methods: the goal attainment method and the nondominated sorting genetic algorithm II. These methodologies are numerically illustrated for a multi-criteria weighted connected directed network. Moreover, the simulated results on similar networks of large order and size are analyzed to show the efficiency of the algorithms.

12 citations


Journal ArticleDOI
TL;DR: It is shown that there does not exist a polynomial time multi-criteria approximation scheme for the resource constrained shortest path problem if the number of weight functions is not a constant, and that this result applies to a broad class of problems, including the multi-dimensional knapsack.
Abstract: In the resource constrained shortest path problem we are given a directed graph along with a source node and a destination node, and each arc has a cost and a vector of weights specifying its requirements from a set of resources with finite budget limits. A minimum cost source-destination path is sought such that the total consumption of the arcs from each resource does not exceed its budget limit. In the case of constant number of weight functions we give a fully polynomial time multi-criteria approximation scheme for the problem which returns a source-destination path of cost at most the optimum, however, the path may slightly violate the budget limits. On the negative side, we show that there does not exist a polynomial time multi-criteria approximation scheme for the problem if the number of weight functions is not a constant. The latter result applies to a broad class of problems as well, including the multi-dimensional knapsack, the multi-budgeted spanning tree, the multi-budgeted matroid basis and the multi-budgeted bipartite perfect matching problems.

9 citations


Journal ArticleDOI
TL;DR: Through the comprehensive analysis of various factors that influence the solution activity, this paper proposes a metric that is used to evaluate the solution robustness of the project scheduling, and this metric is taken as the optimization goal.
Abstract: The research object in this paper is the sub network formed by the predecessor’s affect on the solution activity. This paper is to study three types of influencing factors from the predecessors that lead to the delay of starting time of the solution activity on the longest path, and to analyze the influence degree on the delay of the solution activity’s starting time from different types of factors. On this basis, through the comprehensive analysis of various factors that influence the solution activity, this paper proposes a metric that is used to evaluate the solution robustness of the project scheduling, and this metric is taken as the optimization goal. This paper also adopts the iterative process to design a scattered buffer heuristics algorithm based on the robust scheduling of the time buffer. At the same time, the resource flow network is introduced in this algorithm, using the tabu search algorithm to solve baseline scheduling. For the generation of resource flow network in the baseline scheduling, this algorithm designs a resource allocation algorithm with the maximum use of the precedence relations. Finally, the algorithm proposed in this paper and some other algorithms in previous literature are taken into the simulation experiment; under the comparative analysis, the experimental results show that the algorithm proposed in this paper is reasonable and feasible.

8 citations


Posted Content
TL;DR: This paper attempts to address the issue of dynamic programming in the Massively Parallel Computations (MPC) model which is a popular abstraction of MapReduce-like paradigms, and introduces two classes of graph problems that admit dynamic programming solutions on trees.
Abstract: Dynamic programming is a powerful technique that is, unfortunately, often inherently sequential. That is, there exists no unified method to parallelize algorithms that use dynamic programming. In this paper, we attempt to address this issue in the Massively Parallel Computations (MPC) model which is a popular abstraction of MapReduce-like paradigms. Our main result is an algorithmic framework to adapt a large family of dynamic programs defined over trees. We introduce two classes of graph problems that admit dynamic programming solutions on trees. We refer to them as "(polylog)-expressible" and "linear-expressible" problems. We show that both classes can be parallelized in $O(\log n)$ rounds using a sublinear number of machines and a sublinear memory per machine. To achieve this result, we introduce a series of techniques that can be plugged together. To illustrate the generality of our framework, we implement in $O(\log n)$ rounds of MPC, the dynamic programming solution of graph problems such as minimum bisection, $k$-spanning tree, maximum independent set, longest path, etc., when the input graph is a tree.

8 citations


Journal ArticleDOI
TL;DR: This paper proposed a generalised Diskrtra's algorithm for finding shortest paths from a source node to every other node in a fuzzy graph where the arc lengths are represented by some natural words taken from natural language.
Abstract: Computing with words is a soft computing technique to solve the decision-making problem with the information described in natural language. In this paper, we proposed a generalised Diskrtra's algorithm for finding shortest paths from a source node to every other node in a fuzzy graph where the arc lengths are represented by some natural words taken from natural language. Human being generally describes the arc lengths of any shortest path problem in real life by the terms small, large, some, etc. term which do not supply any natural numbers or fuzzy numbers. We describe those terms as words. Here, we use interval type 2 fuzzy set (IT2FS) to capture the uncertainty of the words. A perceptual computer model is introduced to determine the rank of the shortest path which is a collection of words. A numerical example of transportation network is used to illustrate the effectiveness of the proposed method.

8 citations


Journal ArticleDOI
TL;DR: It is shown that several results concerning the length of the longest path/cycle of a random graph naturally translate to $G_p$ if $G$ is an arbitrary graph of minimum degree at least $n-1 and asymptotically best-possible.
Abstract: For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the $G(n,p)$-model. We show that several results concerning the length of the longest path/cycle naturally translate to $G_p$ if $G$ is an arbitrary graph of minimum degree at least $n-1$. For a constant $c>0$ and $p=\frac{c}{n}$, we show that asymptotically almost surely the length of the longest path in $G_p$ is at least $(1-(1+\epsilon(c))ce^{-c})n$ for some function $\epsilon(c)\to 0$ as $c\to \infty$, and the length of the longest cycle is a least $(1-O(c^{- \frac{1}{5}}))n$. The first result is asymptotically best-possible. This extends several known results on the length of the longest path/cycle of a random graph in the $G(n,p)$-model to the random graph model $G_p$ where $G$ is a graph of minimum degree at least $n-1$.

Posted Content
TL;DR: In this paper, the authors proposed a deterministic algorithm to find a k-path in a directed graph that is guaranteed to have few k vertices in an algebraic circuit with a multiplicative error of 1 \pm \varepsilon.
Abstract: We devise an algorithm that approximately computes the number of paths of length $k$ in a given directed graph with $n$ vertices up to a multiplicative error of $1 \pm \varepsilon$. Our algorithm runs in time $\varepsilon^{-2} 4^k(n+m) \operatorname{poly}(k)$. The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic $2^{k}\cdot\operatorname{poly}(n)$ time algorithm to find a $k$-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect $k$-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.

Journal ArticleDOI
TL;DR: The current work can be treated as a support in choosing an appropriate combinatorial model, resulting in polynomial time solution of problems related to searching for the Hamiltonian cycle or path, which are strongly NP-hard in general.

Posted Content
TL;DR: DLGP can effectively generate timing-correct circuit solutions for Single Flux Quantum logic, which is a magnetic-pulse-based, gate-level pipelined superconductive computing fabric and Experimental results confirm that DLGP generates circuits with considerably lower path balancing overheads compared with a baseline full-path-balancing approach.
Abstract: In this paper, a new graph partitioning problem is introduced. The depth of each part is constrained, i.e., the node count in the longest path of the corresponding sub-graph is no more than a predetermined positive integer value p. An additional constraint is enforced such that each part contains only nodes selected from consecutive levels in the graph. The problem is therefore transformed into a Depth-bounded Levelized Graph Partitioning (DLGP) problem, which is solved optimally using a dynamic programming algorithm. As an example application, we have shown that DLGP can effectively generate timing-correct circuit solutions for Single Flux Quantum (SFQ) logic, which is a magnetic-pulse-based, gate-level pipelined superconductive computing fabric. Experimental results confirm that DLGP generates circuits with considerably lower path balancing overheads compared with a baseline full-path-balancing approach. For example, the balancing overhead (a critical measure of quality metric) for the SFQ circuit realization in terms of D-Flip-Flop count is reduced by 3.61 times on average for 10 benchmark circuit, given p=5.

Proceedings ArticleDOI
01 Jan 2018
TL;DR: An algorithmic framework to adapt a large family of dynamic programs on the Massively Parallel Communications model, which is a popular theoretical model of MapReduce-like systems, and shows that both classes of dynamic programming problems can be solved efficiently using a sub linear number of machines and a sublinear memory per machine.
Abstract: Solving large-scale graph problems is a fundamental task in many real-world applications, and it is an increasingly important problem in data analysis. Despite the large effort in designing scalable graph algorithms, many classic graph problems lack algorithms that require only a sublinear number of machines and space in the input size. Specifically when the input graph is large and sparse, which is indeed the case for many real-world graphs, it becomes impossible to store and access all the vertices in one machine - something that is often taken for granted in designing algorithms for massive graphs. The theoretical model that we consider is the Massively Parallel Communications (MPC) model which is a popular theoretical model of MapReduce-like systems. In this paper, we give an algorithmic framework to adapt a large family of dynamic programs on MPC. We start by introducing two classes of dynamic programming problems, namely "(poly log)-expressible" and "linear-expressible" problems. We show that both classes can be solved efficiently using a sublinear number of machines and a sublinear memory per machine. To achieve this result, we introduce a series of techniques that can be plugged together. To illustrate the generality of our framework, we implement in O(log n) rounds of MPC, the dynamic programming solution of fundamental problems such as minimum bisection, k-spanning tree, maximum independent set, longest path, etc., when the input graph is a tree.

Journal ArticleDOI
TL;DR: Almost hypotraceable graphs, which constitute the extremal case t G = 1, are studied, structural properties of these graphs are given, construction methods for connectivities 1 through 4 are established, and there exists a cubic 3-connected such graph of order 28.

Journal ArticleDOI
TL;DR: A variant of the well-known, NP-complete problem of minimum cut linear arrangement for directed acyclic graphs, where the vertices and edges have weights, is considered, and the aim is to minimize the maximum weight of cut edges in addition to the weight of the last vertex before the cut.

Proceedings ArticleDOI
12 Jun 2018
TL;DR: This work proposes--and proves formally--three generic, low-complexity deadlock avoidance mechanisms that only require local information, which are topology- and routing-independent and their virtual channel count is bounded by the length of the longest path.
Abstract: Recently, the use of graph-based network topologies has been proposed as an alternative to traditional networks such as tori or fat-trees due to their very good topological characteristics. However they pose practical implementation challenges such as the lack of deadlock avoidance strategies. Previous proposals either lack flexibility, underutilise network resources or are exceedingly complex. We propose--and prove formally--three generic, low-complexity deadlock avoidance mechanisms that only require local information. Our methods are topology- and routing-independent and their virtual channel count is bounded by the length of the longest path. We evaluate our algorithms through an extensive simulation study to measure the impact on the performance using both synthetic and realistic traffic. First we compare against a well-known HPC mechanism for dragonfly and achieve similar performance level. Then we moved to Graph-based networks and show that our mechanisms can greatly outperform traditional, spanning-tree based mechanisms, even if these use a much larger number of virtual channels. Overall, our proposal provides a simple, flexible and high performance deadlock-avoidance solution.

Patent
28 May 2018
TL;DR: In this paper, a signal transmission method for a plug-and-play quantum cryptography system including a first communication device and a second communication device is described. But the method is performed by a plug and play system.
Abstract: The present invention relates to a signal transmitting method, which is performed by a plug and play quantum cryptography system including a first communication device and a second communication device, comprising: step (a) of transmitting quantum signals generated by the first communication device to the second communication device by being divided into a first path and a second path which are different from each other, and transmitting a first quantum signal through the first path which is long and a second quantum signal through the second path which is short; and step (b) of receiving the first quantum signal and the second quantum signal transmitted from the second communication device, and changing arrival times of quantum signals 1-1 and 1-2 and quantum signals 2-1 and 2-2 by being divided into the first path and the second path, wherein the arrival time of the quantum signal 2-2 which passes the shortest path among the quantum signals is delayed to generate interference with the quantum signal 1-1 which passes the longest path among the quantum signals.

Journal ArticleDOI
01 Feb 2018
TL;DR: In this paper, an optimization-based approach for finding the genome sequence as a longest sequence that is consistent with the given contig and linkage information is proposed. But this approach does not address the problem of constructing a set of disjoint paths, which would require additional steps of gap filling and scaffold extension, involving additional work.
Abstract: This work focuses simultaneously on both the scaffolding and gap filling phases of de nouveau genome assembly. Given a set of contigs and their relationships--overlaps and/or remoteness in terms of distances between them--we propose an optimization-based approach for finding the genome sequence as a longest sequence that is consistent with the given contig and linkage information. Specifically, we define a graph, which we call a contig graph, that encodes information about contigs and overlaps and mate-pair distances between them, and reduce the scaffolding problem to the problem of finding a longest simple path in that graph such that as many as possible mate-pairs distances are satisfied. Since both conditions cannot generally be simultaneously satisfied, our objective function is a linear combination of the path length and penalties for distance mismatches. ​​Unlike the shortest path problem with non-negative weights, for which efficient polynomial-time algorithms exist, the longest weighted path problem is NP-hard. We solve this problem by reformulating it as a mixed integer linear program (MILP) and develop a method that exactly solves the resulting program on genomes of up to 165 contigs and up to 6682 binary variables. An advantage of our approach is that the modeling of scaffolding as a longest path problem allows one to solve simultaneously several subtasks specific for this problem like: contig orientation and ordering, repeats, gap filling, and scaffold extension, which in other approaches are targeted as separate problems. We are not aware of previous approaches on scaffolding based on the longest path problem reduction. A drawback of the typically used strategy of constructing a set of disjoint paths, rather than a single path, is that it would require additional steps of gap filling and scaffold extension, involving additional work. Moreover, it would make impossible to find a provably optimal final solution, since, even if each separate problem is implemented optimally, their combination may not be optimal. We tested this model on a set of chloroplast and bacteria genome data and showed that it allows to assemble the complete genome as a single scaffold. Compared to the publicly available scaffolding tools that we have tested, our solution produces assemblies of significantly higher quality.

Proceedings ArticleDOI
01 Oct 2018
TL;DR: By estimating spatial orientation of the agents with single antenna, the accuracy is improved by 96% over crowdsourcing only and the scheduling problem is solved with a 3-approximation ratio in polynomial time that exhibits statistically similar performance under variety of urban locale across multiple continents.
Abstract: The distributed nature of policy violations in spectrum sharing necessitate the use of mobile autonomous agents (e.g., UAVs, self-driving cars, crowdsourcing) to implement cost-effective enforcement systems. We define this problem as Multi-agent 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 1.2km per unit diagonal length of a rectangular geographic area, even when there are twice as many infractions as agents.

Posted Content
TL;DR: In this article, it was shown that in such a partition, the longest path can have length asymptotically $N^{1-o(1) + 1/n 2/n 1/1/n/n log n/n) in the divisor graph, where n is the number of paths in the graph.
Abstract: It is known that the longest simple path in the divisor graph that uses integers $\leq N$ is of length $\asymp N/\log N$. We study the partitions of $\{1,2,\dots, N\}$ into a minimal number of paths of the divisor graph, and we show that in such a partition, the longest path can have length asymptotically $N^{1-o(1)}$.

Book ChapterDOI
18 Jun 2018
TL;DR: The Wrapping Layered Graphs Problem (WLGP), which seeks for cut indices that split a given layering into chunks that are drawn side-by-side with a preferably small number of edges wrapping backwards, allows an improved presentation of narrow graphs.
Abstract: We present additions to the widely-used layout method for directed acyclic graphs of Sugiyama et al. that allow to better utilize a prescribed drawing area. The method itself partitions the graph’s nodes into layers. When drawing from top to bottom, the number of layers directly impacts the height of a resulting drawing and is bound from below by the graph’s longest path. As a consequence, the drawings of certain graphs are significantly taller than wide, making it hard to properly display them on a medium such as a computer screen without scaling the graph’s elements down to illegibility. We address this with the Wrapping Layered Graphs Problem (WLGP), which seeks for cut indices that split a given layering into chunks that are drawn side-by-side with a preferably small number of edges wrapping backwards. Our experience and a quantitative evaluation indicate that the proposed wrapping allows an improved presentation of narrow graphs, which occur frequently in practice and of which the internal compiler representation SCG is one example.

Dissertation
01 Jan 2018
TL;DR: Thesis: S.M.M., Massachusetts Institute of Technology, Department of Mechanical Engineering, in conjunction with the Leaders for Global Operations Program at MIT, 2018.
Abstract: Thesis: S.M., Massachusetts Institute of Technology, Department of Mechanical Engineering, in conjunction with the Leaders for Global Operations Program at MIT, 2018.

Journal ArticleDOI
TL;DR: A dynamic programming based algorithm and heuristic are developed to solve the average longest path problem in O(nm) and approximate the average maximum cost flow for directed acyclic graphs (DAG).

Posted Content
TL;DR: In this article, the authors prove two dichotomy results for detecting long paths as patterns in a given graph: the first is to determine the longest induced path in a graph, and the second is to find the longest path to which a graph can be contracted to.
Abstract: We prove two dichotomy results for detecting long paths as patterns in a given graph. The NP-hard problem Longest Induced Path is to determine the longest induced path in a graph. The NP-hard problem Longest Path Contractibility is to determine the longest path to which a graph can be contracted to. By combining known results with new results we completely classify the computational complexity of both problems for $H$-free graphs. Our main focus is on the second problem, for which we design a general contractibility technique that enables us to reduce the problem to a matching problem.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the problem of finding a simple path between two given vertices in an arc weighted directed multigraph such that the path length is equal to a given number or it does not fall into the given forbidden intervals (gaps).
Abstract: We study new decision and optimization problems of finding a simple path between two given vertices in an arc weighted directed multigraph such that the path length is equal to a given number or it does not fall into the given forbidden intervals (gaps). A fairly complete computational complexity classification is provided and exact and approximation algorithms are suggested.

Journal ArticleDOI
TL;DR: The family of Directed Acyclic Graphs as well as some related graphs are analyzed with respect to extremal behavior in relation with the family of intersection graphs for families of boxes with transverse intersection.
Abstract: This paper studies the maximum number of edges of a Directed Acyclic Graph (DAG) with n vertices in terms of it’s longest path l . We prove that in general this number is the Turan number t ( n , l + 1) , the maximum number of edges in a graph with n vertices without a clique of size l + 2 . Furthermore, we find the maximum number of edges in a DAG which is either reduced, strongly reduced or extremely reduced and we relate this extremal result with the family of intersection graphs of families of boxes with transverse intersection.


Posted Content
TL;DR: This paper presents 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 this framework yields polynomial-time algorithms for the three problems.
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 the complexity of some constrained vertex separators are polynomial-time solvable in cographs. We propose polynomial-time algorithms for 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, our framework yields polynomial-time algorithms for all three problems.