Showing papers on "Longest path problem published in 2017"

Lossy kernelization

Abstract: In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity. However, as opposed to the original notion of kernelization, our definitions com- bine well with approximation algorithms and heuristics. The key new definition is that of a polynomial size α-approximate kernel. Loosely speaking, a polynomial size α-approximate kernel is a polynomial time pre-processing algorithm that takes as input an instance (I, k) to a parameterized problem, and outputs another instance (I′,k′) to the same problem, such that |I′| + k′ ≤ kO(1). Additionally, for every c ≥ 1, a c-approximate solution s′ to the pre-processed instance (I′, k′) can be turned in polynomial time into a (c · α)-approximate solution s to the original instance (I,k). Amongst our main technical contributions are α-approximate kernels of polynomial size for three problems, namely Connected Vertex Cover, Disjoint Cycle Packing and Disjoint Factors. These problems are known not to admit any polynomial size kernels unless NP ⊆ coNP/Poly. Our approximate kernels simultaneously beat both the lower bounds on the (normal) kernel size, and the hardness of approximation lower bounds for all three problems. On the negative side we prove that Longest Path parameterized by the length of the path and Set Cover parameterized by the universe size do not admit even an α-approximate kernel of polynomial size, for any α≥1, unless NP ⊆ coNP/Poly. In order to prove this lower bound we need to combine in a non-trivial way the techniques used for showing kernelization lower bounds with the methods for showing hardness of approximation.

Intractability of Time-Optimal Multirobot Path Planning on 2D Grid Graphs with Holes

Abstract: The most tight intractability results for graph-based Multirobot Path Planning (MPP), proven recently, state that time-optimal and distance-optimal MPP problems are NP-hard on planar graphs. In this letter, we go one step further for what concerns the time-optimal objectives, and prove that such problems remain NP-hard when restricting the planar graph to a 2D grid graph with holes, which is a discretization widely used in robotics. Our reduction (from the Boolean satisfiability problem) cannot be easily modified for the distance-optimal objectives, whose hardness remains an open problem.

Multiple-Source Multiple-Sink Maximum Flow in Directed Planar Graphs in Near-Linear Time

Abstract: We give an $O(n \log^3 n)$ algorithm that, given an $n$-node directed planar graph with arc capacities, a set of source nodes, and a set of sink nodes finds a maximum flow from the sources to the sinks. Previously, the fastest algorithms known for this problem were those for general graphs.

Shortest path problem on single valued neutrosophic graphs

Abstract: A single valued neutrosophic graph is a generalized structure of fuzzy graph, intuitionistic fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs and intuitionistic fuzzy graphs. This paper addresses for the first time, the shortest path in an acyclic neutrosophic directed graph using ranking function. Here each edge length is assigned to single valued neutrosophic numbers instead of a real number. The neutrosophic number is able to represent the indeterminacy in the edge (arc) costs of neutrosophic graph. A proposed algorithm gives the shortest path and shortest path length from source node to destination node. Finally an illustrative example also included to demonstrate the proposed method in solving path problems with single valued neutrosophic arcs.

A type of biased consensus-based distributed neural network for path planning

Abstract: In this paper, a unified scheme is proposed for solving the classical shortest path problem and the generalized shortest path problem, which are highly nonlinear. Particularly, the generalized shortest path problem is more complex than the classical shortest path problem since it requires finding a shortest path among the paths from a vertex to all the feasible destination vertices. Different from existing results, inspired by the optimality principle of Bellman’s dynamic programming, we formulate the two types of shortest path problems as linear programs with the decision variables denoting the lengths of possible paths. Then, biased consensus neural networks are adopted to solve the corresponding linear programs in an efficient and distributed manner. Theoretical analysis guarantees the performance of the proposed scheme. In addition, two illustrative examples are presented to validate the efficacy of the proposed scheme and the theoretical results. Moreover, an application to mobile robot navigation in a maze further substantiates the efficacy of the proposed scheme.

Towards shortest path computation using Dijkstra algorithm

Abstract: Shortest Path issue plays an important role in applications of road network such as handling city emergency way and guiding driver system. The concepts of network analysis with traffic issues are recognized. The condition of traffic among a city changes periodically and there are usually large amounts of requests occur, it needs to be solve quickly. By using the Dijkstra's Algorithm, the above problems can be solved through shortest paths. The shortest path and the best path is computed based on the problem of traffic condition shortest path. This plays an important role in navigation systems as it can help to make sensible decision and time saving decisions. The main purpose is to get the implementation at low cost. Thus, it brings a new framework called towards shortest path which enables drivers to quickly and effectively collect the shortest path as well as alternative path and traffic information. An impressive the result is that the driver can get their shortest path result and also gives alternative paths for the same route with the traffic count. Our experimental resultsfind that it is better invarious parameters and it gives relatively fast response time, for shortest path problem.

An Indexing Framework for Queries on Probabilistic Graphs

Abstract: Information in many applications, such as mobile wireless systems, social networks, and road networks, is captured by graphs. In many cases, such information is uncertain. We study the problem of querying a probabilistic graph, in which vertices are connected to each other probabilistically. In particular, we examine “source-to-target” queries (ST-queries), such as computing the shortest path between two vertices. The major difference with the deterministic setting is that query answers are enriched with probabilistic annotations. Evaluating ST-queries over probabilistic graphs is nP-hard, as it requires examining an exponential number of “possible worlds”—database instances generated from the probabilistic graph. Existing solutions to the ST-query problem, which sample possible worlds, have two downsides: (i) a possible world can be very large and (ii) many samples are needed for reasonable accuracy. To tackle these issues, we study the ProbTree, a data structure that stores a succinct, or indexed, version of the possible worlds of the graph. Existing ST-query solutions are executed on top of this structure, with the number of samples and sizes of the possible worlds reduced. We examine lossless and lossy methods for generating the ProbTree, which reflect the tradeoff between the accuracy and efficiency of query evaluation. We analyze the correctness and complexity of these approaches. Our extensive experiments on real datasets show that the ProbTree is fast to generate and small in size. It also enhances the accuracy and efficiency of existing ST-query algorithms significantly.

Aligning sequences to general graphs in O(V + mE) time

Abstract: Graphs are commonly used to represent sets of sequences. Either edges or nodes can be labeled by sequences, so that each path in the graph spells a concatenated sequence. Examples include graphs to represent genome assemblies, such as string graphs and de Bruijn graphs, and graphs to represent a pan-genome and hence the genetic variation present in a population. Being able to align sequencing reads to such graphs is a key step for many analyses and its applications include genome assembly, read error correction, and variant calling with respect to a variation graph. Given the wide range of applications of this basic problem, it is surprising that algorithms with optimal runtime are, to the best of our knowledge, yet unknown. In particular, aligning sequences to cyclic graphs currently represents a challenge both in theory and practice. Here, we introduce an algorithm to compute the minimum edit distance of a sequence of length m to any path in a node-labeled directed graph (V,E) in O(V+m|E|) time and O(|V|) space. The corresponding alignment can be obtained in the same runtime using O(√m|V|) space. The time complexity depends only on the length of the sequence and the size of the graph. In particular, it does not depend on the cyclicity of the graph, or any other topological features.

Acyclic Partitioning of Large Directed Acyclic Graphs

Abstract: Finding a good partition of a computational directed acyclic graph associated with an algorithm can help find an execution pattern improving data locality, conduct an analysis of data movement, and expose parallel steps. The partition is required to be acyclic, i.e., the inter-part edges between the vertices from different parts should preserve an acyclic dependency structure among the parts. In this work, we adopt the multilevel approach with coarsening, initial partitioning, and refinement phases for acyclic partitioning of directed acyclic graphs and develop a direct k-way partitioning scheme. To the best of our knowledge, no such scheme exists in the literature. To ensure the acyclicity of the partition at all times, we propose novel and efficient coarsening and refinement heuristics. The quality of the computed acyclic partitions is assessed by computing the edge cut, the total volume of communication between the parts, and the critical path latencies. We use the solution returned by well-known undirected graph partitioners as a baseline to evaluate our acyclic partitioner, knowing that the space of solution is more restricted in our problem. The experiments are run on large graphs arising from linear algebra applications.

On succinct representation of directed graphs

Abstract: Directed graphs encode meaningful dependencies among objects ubiquitously. This paper introduces new and simple representations for labeled directed graphs with the properties of being succinct (space is information-theoretically optimal); in which we avoid exploiting a-priori knowledge on digraph regularity such as triangularity, separability, planarity, symmetry and sparsity. Our results have direct implications to model directed graphs by using single integer numbers effectively, which is significant to enable canonical (generation of graph instances is unique) and efficient (coding and decoding take polynomial time) encodings for learning and optimization algorithms. To the best of our knowledge, the proposed representations are the first known in the literature.

An extension principle based solution approach for shortest path problem with fuzzy arc lengths

Abstract: A shortest path problem on a network in the presence of fuzzy arc lengths is focused in this paper. The aim is to introduce the shortest path connecting the first and last vertices of the network which has minimum fuzzy sum of arc lengths among all possible paths. In this study a solution algorithm based on the extension principle of Zadeh is developed to solve the problem. The algorithm decomposes the fuzzy shortest path problem into two lower bound and upper bound sub-problems. Each sub-problem is solved individually in different $$\alpha$$ levels to obtain the shortest path, its fuzzy length and its associated membership function value. The proposed method contains no fuzzy ranking function and also for each $$\alpha$$ -cut, it gives a unique lower and upper bound for the fuzzy length of the shortest path. The algorithm is examined over some well-known networks from the literature and its performance is superior to the existent methods.

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

Abstract: We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of directed sparse graphs. While the parameterized complexity of Steiner Tree parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W2-hard on general graphs, and hence unlikely to be fixed parameter tractable (FPT). The undirected Steiner Tree problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs.

Optimal Path Planning using Equilateral Spaces Oriented Visibility Graph Method

Abstract: Path planning has been an important aspect in the development of autonomous cars in which path planning is used to find a collision-free path for the car to traverse from a starting point Sp to a target point Tp The main criteria for a good path planning algorithm include the capability of producing the shortest path with a low computation time Low computation time makes the autonomous car able to re-plan a new collision-free path to avoid accident However, the main problem with most path planning methods is their computation time increases as the number of obstacles in the environment increases In this paper, an algorithm based on visibility graph (VG) is proposed In the proposed algorithm, which is called Equilateral Space Oriented Visibility Graph (ESOVG), the number of obstacles considered for path planning is reduced by introducing a space in which the obstacles lie This means the obstacles located outside the space are ignored for path planning From simulation, the proposed algorithm has an improvement rate of up to 90% when compared to VG This makes the algorithm is suitable to be applied in real-time and will greatly accelerate the development of autonomous cars in the near future

On the complexity of minimum-link path problems

Abstract: We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the vertices and/or the edges of the path are restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2 dimensions, and provide first results in dimensions 3 and higher for several variants of the problem. Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [Ghosh et al.'2012] despite a large body of work on the topic. We also resolve the open problem from [Mitchell et al.'1992] mentioned in the handbook [Goodman and Rourke'2004] (see Chapter 27.5, Open problem 3) and The Open Problems Project [http://maven.smith.edu/~orourke/TOPP/] (see Problem 22): "What is the complexity of the minimum-link path problem in 3-space?" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.

Finding the Shortest Path in Stochastic Graphs Using Learning Automata and Adaptive Stochastic Petri Nets

Abstract: Shortest path problem in stochastic graphs has been recently studied in the literature and a number of algorithms has been provided to find it using varieties of learning automata models However, all these algorithms suffer from two common drawbacks: low speed and lack of a clear termination condition In this paper, we propose a novel learning automata-based algorithm for this problem which can speed up the process of finding the shortest path using parallelism For this parallelism, several traverses are initiated, in parallel, from the source node towards the destination node in the graph During each traverse, required times for traversing from the source node up to any visited node are estimated The time estimation at each visited node is then given to the learning automaton residing in that node Using different time estimations provided by different traverses, this learning automaton gradually learns which neighbor of the node is on the shortest path To set a condition for the termination of the

Secluded Connectivity Problems

Abstract: Consider a setting where possibly sensitive information sent over a path in a network is visible to every neighbor of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path P can be measured as the number of nodes adjacent to it, denoted by N[P]. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected n-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $$O(2^{\log ^{1-\epsilon }n})$$ for any $$\epsilon >0$$ (under an appropriate complexity assumption), but is approximable with ratio $$\sqrt{\Delta }+3$$ , where $$\Delta $$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs. Turning to the more general problem of finding a minimum exposure Steiner tree connecting a given set of k terminals, the picture becomes more involved. In undirected unweighted graphs with unbounded degree, we present an approximation algorithm with ratio $$\min \{\Delta , n/k, \sqrt{2n},O(\log k \cdot (k+\sqrt{\Delta }))\}$$ . On unweighted undirected bounded-degree graphs, the problem is still polynomial when the number of terminals is fixed, but if the number of terminals is arbitrary, then the problem becomes NP-hard again.

Efficient computation of distance labeling for decremental updates in large dynamic graphs

Abstract: Since today's real-world graphs, such as social network graphs, are evolving all the time, it is of great importance to perform graph computations and analysis in these dynamic graphs. Due to the fact that many applications such as social network link analysis with the existence of inactive users need to handle failed links or nodes, decremental computation and maintenance for graphs is considered a challenging problem. Shortest path computation is one of the most fundamental operations for managing and analyzing large graphs. A number of indexing methods have been proposed to answer distance queries in static graphs. Unfortunately, there is little work on answering such queries for dynamic graphs. In this paper, we focus on the problem of computing the shortest path distance in dynamic graphs, particularly on decremental updates (i.e., edge deletions). We propose maintenance algorithms based on distance labeling, which can handle decremental updates efficiently. By exploiting properties of distance labeling in original graphs, we are able to efficiently maintain distance labeling for new graphs. We experimentally evaluate our algorithms using eleven real-world large graphs and confirm the effectiveness and efficiency of our approach. More specifically, our method can speed up index re-computation by up to an order of magnitude compared with the state-of-the-art method, Pruned Landmark Labeling (PLL).

A smooth local path planning algorithm based on modified visibility graph

Abstract: Path planning is an essential and inevitable problem in robotics. Trapping in local minima and discontinuities often exist in local path planning. To overcome these drawbacks, this paper presents a smooth path planning algorithm based on modified visibility graph. This algorithm consists of three steps: (1) polygons are generated from detected obstacles; (2) a collision-free path is found by simultaneous visibility graph construction and path search by A∗ (SVGA); (3) the path is smoothed by B-spline curves and particle swarm optimization (PSO). Simulation experiment results show the effectiveness of this algorithm, and a smooth path can be found fleetly.

Finding Dominating Induced Matchings in $$P_8$$P8-Free Graphs in Polynomial Time

Abstract: Let $$G=(V,E)$$G=(V,E) be a finite undirected graph. An edge set $$E' \subseteq E$$E?⊆E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of $$E'$$E?. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is $${\mathbb {NP}}$$NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three and is solvable in linear time for $$P_7$$P7-free graphs. However, its complexity was open for $$P_k$$Pk-free graphs for any $$k \ge 8$$k?8; $$P_k$$Pk denotes the chordless path with k vertices and $$k-1$$k-1 edges. We show in this paper that the weighted DIM problem is solvable in polynomial time for $$P_8$$P8-free graphs.

Polynomial-time algorithms for the Longest Induced Path and Induced Disjoint Paths problems on graphs of bounded mim-width

Abstract: We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give $n^{\mathcal{O}(w)}$-time algorithms on graphs of mim-width at most $w$, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and $H$-Induced Topological Minor for fixed $H$. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs, Convex graphs, $k$-Trapezoid, Circular $k$-Trapezoid, $k$-Polygon, Dilworth-$k$ and Co-$k$-Degenerate graphs for fixed $k$.

A Global Path Planning Algorithm Based on Bidirectional SVGA

Abstract: For path planning algorithms based on visibility graph, constructing a visibility graph is very time-consuming. To reduce the computing time of visibility graph construction, this paper proposes a novel global path planning algorithm, bidirectional SVGA (simultaneous visibility graph construction and path optimization by A?). This algorithm does not construct a visibility graph before the path optimization. However it constructs a visibility graph and searches for an optimal path at the same time. At each step, a node with the lowest estimation cost is selected to be expanded. According to the status of this node, different through lines are drawn. If this line is free-collision, it is added to the visibility graph. If not, some vertices of obstacles which are passed through by this line are added to the OPEN list for expansion. In the SVGA process, only a few visible edges which are in relation to the optimal path are drawn and the most visible edges are ignored. For taking advantage of multicore processors, this algorithm performs SVGA in parallel from both directions. By SVGA and parallel performance, this algorithm reduces the computing time and space. Simulation experiment results in different environments show that the proposed algorithm improves the time and space efficiency of path planning.