Showing papers on "Longest path problem published in 1997"

On approximating the longest path in a graph

Abstract: We consider the problem of approximating the longest path in undirected graphs. In an attempt to pin down the best achievable performance ratio of an approximation algorithm for this problem, we present both positive and negative results. First, a simple greedy algorithm is shown to find long paths in dense graphs. We then consider the problem of finding paths in graphs that are guaranteed to have extremely long paths. We devise an algorithm that finds paths of a logarithmic length in Hamiltonian graphs. This algorithm works for a much larger class of graphs (weakly Hamiltonian), where the result is the best possible. Since the hard case appears to be that of sparse graphs, we also consider sparse random graphs. Here we show that a relatively long path can be obtained, thereby partially answering an open problem of Broderet al. To explain the difficulty of obtaining better approximations, we also prove hardness results. We show that, for any ź 0, finding an approximation of rationź is alsoNP-hard. As evidence toward this conjecture, we show that if any polynomial-time algorithm can approximate the longest path to a ratio of $$2^{O(\log ^{1 - \varepsilon } n)} $$ , for any ź>0, thenNP has a quasi-polynomial deterministic time simulation. The hardness results apply even to the special case where the input consists of bounded degree graphs.

Genetic algorithms for solving shortest path problems

Abstract: In this study, we investigated the possibility of using genetic algorithms to solve shortest path problems. The most thorny and critical task for developing a genetic algorithm to this problem is how to encode a path in a graph into a chromosome. A priority-based encoding method is proposed which can potentially represent all possible paths in a graph. Because a variety of network optimization problems may be solved, either exactly or approximately, by identifying shortest path, this studies will provide a base for constructing efficient solution procedures for shortest path-based network optimization problems. The proposed approach has been tested on three randomly generated problems with different size from 6 nodes to 70 nodes and from 10 edges to 211 edges. The experiment results are very encouraging: it can find the known optimum very rapidly with very high probability. It can be believed that genetic algorithms may hopefully be a new approach for such kinds of difficult-to-solve problems.

A framed-quadtree approach for determining Euclidean shortest paths in a 2-D environment

Abstract: In this paper we investigate the problem of finding a Euclidean (L/sub 2/) shortest path between two distinct locations in a planar environment. We propose a novel cell decomposition approach which calculates an L/sub 2/ distance transform through the use of a circular path-planning wave. The proposed method is based on a new data structure, called the framed-quadtree, which combines together the accuracy of high resolution grid-based path planning techniques with the efficiency of quadtree-based techniques, hence having the advantages of both. The heart of this method is a linear time algorithm for computing certain special dynamic Voronoi diagrams. The proposed method does not place any unrealistic constraints on obstacles or on the environment and represents an improvement in accuracy and efficiency over traditional path planning approaches in this area.

Recent results on the single-source shortest paths problem

Abstract: We summarize the currently best known theoretical results for the single-source shortest paths problem for directed graphs with non-negative edge weights. We also point out that a recent result due to Cherkassky, Goldberg and Silverstein (1996) leads to even better time bounds for this problem than claimed by the authors.

Semi-Dynamic Shortest Paths and Breadth-First Search in Digraphs

Abstract: We show how to maintain a shortest path tree of a general directed graph G with unit edge weights and n vertices, during a sequence of edge deletions or a sequence of edge insertions, in O(n) amortized time per operation using linear space Distance queries can be answered in constant time, while shortest path queries can be answered in time linear in the length of the retrieved path These results are extended to the case of integer edge weights in [1,C], with a bound of O(Cn) amortized time per operation

Complexity of the agv shortest path and single-loop guide path layout problems

Abstract: The guide path layout is one of the most important variables in the design of automated guided vehicle systems (AGVS). Many alternative guide path configurations have been proposed in previous research. Two of the simplest are the shortest path and shortest single-loop configurations. However, the optimal solutions to these two problems have not been easy to obtain. In this paper, we show why by proving the NP-completeness of these problems using graph theoretic methods.

On calculation of the detour index

Abstract: An algorithm for the detection of the longest path between any two vertices of a graph was proposed. The method was used to calculate the detour index of fused bicyclic structures. Analytical formulas for the detour index of fused bicyclic structures were derived, and it was shown the these can only be third-order polynomials.

Unification of budgeting and placement

Abstract: In this paper we present a novel formulation for thenet-based timing-driven placement problem.The new formulationperforms budgeting (net delay upper bounds) andplacement modification simultaneously thus alleviates theproblem of going back-and-forth between budgeting andplacement.An algorithm to accomplished the proposedtask is presented.The proposed algorithm uses a simulatedannealing approach and a modified graph-based simplexmethod.A general formulation of timing-sriven placementis presented.It is proved that both net-based andpath-based approaches to timing-driven placement are specialcases of a more general formulation.The proposed algorithmhas been incorporated into a (timing-driven) placementpackage.Experiments on MCNC benchmarks showstrong results.The proposed algorithm offers 54% to 68%reduction over the longest path compared with the existingalgorithms.

Realizing Interval Graphs with Size and Distance Constraints

Abstract: We study the following problem: given an interval graph, does it have a realization which satisfies additional constraints on the distances between interval endpoints? This problem arises in numerous applications in which topological information on intersection of pairs of intervals is accompanied by additional metric information on their order, distance, or size. An important application is physical mapping, a central challenge in the human genome project. Our results are (1) a polynomial algorithm for the problem on interval graphs which admit a unique clique order (UCO graphs). This class of graphs properly contains all prime interval graphs. (2) In case all constraints are upper and lower bounds on individual interval lengths, the problem on UCO graphs is linearly equivalent to deciding if a system of difference inequalities is feasible. (3) Even if all the constraints are prescribed lengths of individual intervals, the problem is NP-complete. Hence, problems (1) and (2) are also NP-complete on arbitrary interval graphs.

An Approximation Algorithm for the Bandwidth Problem on Dense Graphs

Abstract: The bandwidth problem is the problem of numbering the vertices of a given graph G such that the maximum difference between two numbers of adjacent vertices is minimal. The problem is known to be NP-complete and there are only few algorithms for rather special cases of the problem [HaMaMo91] [Kra87] [Sax80] [Smi95]. In this paper we present a randomized 3-approximation algorithm for the bandwidth problem restricted to dense graphs and a randomized 2-approximation algorithm for the same problem on directed dense graphs.

A lower bound for interval routing in general networks

Abstract: Interval routing is a space-efficient routing method for point-to-point communication networks. The method has drawn considerable attention in recent years because of its being incorporated into the design of a commercially available routing chip. The method is based on proper labeling of edges of the graph with intervals. An optimal labeling would result in routing of messages through the shortest paths. Optimal labelings have existed for regular as well as some of the common topologies, but not for arbitrary graphs. In fact, it has already been shown that it is impossible to find optimal labelings for arbitrary graphs. In this paper, we prove a 7D/40 1 lower bound for interval routing in arbitrary graphs, where D is the diameter—i.e., the best any interval labeling scheme could do is to produce a longest path having a length of at least 7D/40 1. q 1997 John Wiley & Sons, Inc.

Two dimensional collision-free path planning using linear parametric curve

Abstract: An effective algorithm for planning a collision-free path based on linear parametric curve is developed. A collision-free path is viewed as a series of segmented polynomial curves in a space that does not interfere with any object in a given work space. It is assumed that the path connecting start and target points has no width. The algorithm presented here uses a linear parametric curve as a base curve and maps objects in Euclidean Space (ES) into objects in Control Point Space (CPS) through intersection checks between path and obstacles. A path having a single control point is investigated here. The Free Space (FS) of CPS identifies a collision-free path in ES, hence any point in the FS of CPS is a candidate for a collision-free path. A CPS completely occupied by obstacle images indicates no collision-free path is available with a single control point and a search based on a multiple control point is required. The shortest path with minimum search time is obtained by setting CPS in elliptic coordinate. Considerations are given to get a path with the smallest maximum curvature by selecting the control point toward the bisection line of ST and smoothing the path. ©1997 John Wiley & Sons, Inc.

A generalized threshold algorithm for the shortest path problem with time windows.

Abstract: In this paper, we present a new labeling algorithm for the shortest path problem with time windows (SPPTW). It is generalized from the threshold algorithm for the unconstrained shortest path problem. Our computational experiments show that this generalized threshold algorithm outperforms a label setting algorithm for the SPPTW on a set of randomly generated test problems. The average running time of the new algorithm is about 40% less than the label setting algorithm, which is today the best algorithm based on published experimental evidence.

Improved approximations for shallow-light spanning trees

Abstract: We consider the bicriteria optimization problem of computing a shallow-light tree. Given a directed graph with two unrelated cost functions defined on its edges: weight and length, and a designated root vertex, the goal is to find a minimum weight spanning tree such that the path lengths from its root to the rest of the vertices are bounded. This problem has several applications in network and VLSI design, and information retrieval. We give a polynomial time algorithm for finding a spanning tree whose weight is O(log |V|) times the weight of an optimal shallow-light tree, where the path lengths from the root to the rest of the vertices are at most twice the given bounds. We extend our technique to handle two variants of the problem: one in which the length bound is given on the average length of a path from the root to a vertex, and another tricriteria budgeted version. Our paper provides the first non-trivial approximation factors for directed graphs, and improves on previous results for undirected graphs.

The disjoint cliques problem

Abstract: Given a graph G = (V, E), we consider the problem of finding a set of D pairwise disjoint cliques in the graph with maximum overall number of vertices. We determine the computational complexity of this problem restricted to a variety of different graph classes. We give polynomial time algorithms for the problem restricted to interval graphs, cographs, directed path graphs and partial k-trees. In contrast, we show the NP-completeness of this problem for undirected path graphs. Moreover, we investigate a closely related scheduling problem. Given D times units, we look for a sequence of workers w 1 ,...,w K and a partition J 1 ,..., J k of the job set such that J i can be executed by w i within D time units. The goal is to find a sequence with minimum total wage of the workers.

An Optimal Lower Bound for Interval Routing in General Networks

Abstract: Interval routing is a space-efficient (compact) routing method for point-topoint communication networks. The method is based on proper labeling of edges of the graph with intervals. An optimal labeling would result in routing of messages through the shortest paths. Optimal labelings exist for regular as well as some of the common topologies, but not for arbitrary graphs. It has been shown that it is impossible to find optimal labelings for arbitrary graphs [4]. In this paper, we prove the lower bound of 2D − 3 on the longest routing path for arbitrary graphs, where D = O( √ n) is the graph’s diameter and n is the number of nodes, as well as a lower bound of 2D − o(D) for D = O(n). Our results are very close to the best known upper bound which is 2D.