Showing papers on "Longest path problem published in 2007"

Graph evolution: Densification and shrinking diameters

Abstract: How do real graphs evolve over timeq What are normal growth patterns in social, technological, and information networksq Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network or in a very small number of snapshots; these include heavy tails for in- and out-degree distributions, communities, small-world phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time.Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time with the number of edges growing superlinearly in the number of nodes. Second, the average distance between nodes often shrinks over time in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) or O(log(log n)).Existing graph generation models do not exhibit these types of behavior even at a qualitative level. We provide a new graph generator, based on a forest fire spreading process that has a simple, intuitive justification, requires very few parameters (like the flammability of nodes), and produces graphs exhibiting the full range of properties observed both in prior work and in the present study.We also notice that the forest fire model exhibits a sharp transition between sparse graphs and graphs that are densifying. Graphs with decreasing distance between the nodes are generated around this transition point.Last, we analyze the connection between the temporal evolution of the degree distribution and densification of a graph. We find that the two are fundamentally related. We also observe that real networks exhibit this type of relation between densification and the degree distribution.

Single-Track Train Timetabling with Guaranteed Optimality: Branch-and-Bound Algorithms with Enhanced Lower Bounds

Abstract: A single-track train timetabling problem is studied in order to minimize the total train travel time, subject to a set of operational and safety requirements. This research proposes a generalized resource-constrained project scheduling formulation which considers segment and station headway capacities as limited resources, and presents a branch-and-bound solution procedure to obtain feasible schedules with guaranteed optimality. The search algorithm chronologically adds precedence relation constraints between conflicting trains to eliminate conflicts, and the resulting sub-problems are solved by the longest path algorithm to determine the earliest start times for each train in different segments. This study adapts three approaches to effectively reduce the solution space. First, a Lagrangian relaxation based lower bound rule is used to dualize segment and station entering headway capacity constraints. Second, an exact lower bound rule is used to estimate the least train delay for resolving the remaining crossing conflicts in a partial schedule. Third, a tight upper bound is constructed by a beam search heuristic method. Comprehensive numerical experiments are conducted to illustrate the computational performance of the proposed lower bound rules and heuristic upper bound construction methods.

The On-Line Shortest Path Problem Under Partial Monitoring

Abstract: The on-line shortest path problem is considered under various models of partial monitoring. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a path between two distinguished vertices such that the loss of the chosen path (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit problem, after choosing a path, the decision maker learns only the weights of those edges that belong to the chosen path. For this problem, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best path, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1/√n and depends only polynomially on the number of edges of the graph. The algorithm can be implemented with complexity that is linear in the number of rounds n (i.e., the average complexity per round is constant) and in the number of edges. An extension to the so-called label efficient setting is also given, in which the decision maker is informed about the weights of the edges corresponding to the chosen path at a total of m ≪ n time instances. Another extension is shown where the decision maker competes against a time-varying path, a generalization of the problem of tracking the best expert. A version of the multi-armed bandit setting for shortest path is also discussed where the decision maker learns only the total weight of the chosen path but not the weights of the individual edges on the path. Applications to routing in packet switched networks along with simulation results are also presented.

An experimental study of a parallel shortest path algorithm for solving large-scale graph instances

Abstract: We present an experimental study of the single source shortest path problem with non-negative edge weights (NSSP) on large-scale graphs using the Δ-stepping parallel algorithm. We report performance results on the Cray MTA-2, a multithreaded parallel computer. The MTA-2 is a high-end shared memory system offering two unique features that aid the efficient parallel implementation of irregular algorithms: the ability to exploit fine-grained parallelism, and low-overhead synchronization primitives. Our implementation exhibits remarkable parallel speedup when compared with competitive sequential algorithms, for low-diameter sparse graphs. For instance, Δ-stepping on a directed scale-free graph of 100 million vertices and 1 billion edges takes less than ten seconds on 40 processors of the MTA-2, with a relative speedup of close to 30. To our knowledge, these are the first performance results of a shortest path problem on realistic graph instances in the order of billions of vertices and edges.

Partitioning and modularity of graphs with arbitrary degree distribution.

Abstract: We solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut size to scale universally with k. In contrast, earlier results studying the problem in graphs with a Poissonian degree distribution had found a scaling with kFu and Anderson, J. Phys. A 19, 1605 1986. Our results also generalize to the problem of q partitioning. They can be used to find the expected modularity Q Newman and Girvan, Phys. Rev. E 69, 026113 2004 of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms. Given a graph or network GN ,M of N nodes and M edges, the problem of graph partitioning is finding a partition of the nodes into q equal sized parts, such that the number of edges connecting different parts, the cut size, is minimal. The solution of this problem has many important practical applications in multiprocessor scheduling for parallel computing, very large-scale integrated chip design VLSI, and data mining 1. Due to the problem being NP-complete 2, i.e., no algorithm is known which is guaranteed to find an optimal solution in a number of steps that grows only polynomially with the size of the graph, only heuristics exist to approximate a solution. Furthermore, one cannot determine in polynomial time whether a solution found is indeed optimal. The only way to assess the quality of a heuristic algorithm or a particular result is to compare with rigorous bounds on the cut size.

Leavitt path algebras and direct limits

Abstract: An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplicity, characterizations of the exchange property, and cancellation conditions for the K-theoretic monoid of equivalence classes of idempotent matrices.

An Improved Solution Algorithm for the Constrained Shortest Path Problem

Abstract: The shortest path problem is one of the classic network problems. The objective of this problem is to identify the least cost path through a network from a pre-determined starting node to a pre-determined terminus node. It has many practical applications and can be solved optimally via efficient algorithms. Numerous modifications of the problem exist. In general, these are more difficult to solve. One of these modified versions includes an additional constraint that establishes an upper limit on the sum of some other arc cost (e.g., travel time) for the path. In this paper, a new optimal algorithm for this constrained shortest path problem is introduced. Extensive computational tests are presented which compare the algorithm to the two most commonly used algorithms to solve it. The results indicate that the new algorithm can solve optimally very large problem instances and is generally superior to the previous ones in terms of solution time and computer memory requirements. This is particularly true for the problem instances that are most difficult to solve. That is, those on large networks and/or where the additional constraint is most constraining.

Hardness of robust network design

Abstract: The authors settle the complexity status of the robust network design problem in undirected graphs. The fact that the flow-cut gap in general graphs can be large, poses some difficulty in establishing a hardness result. Instead, the authors introduce a single-source version of the problem where the flow-cut gap is known to be one. They then show that this restricted problem is coNP-Hard. This version also captures, as special cases, the fractional relaxations of several problems including the spanning tree problem, the Steiner tree problem, and the shortest path problem. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 50–54 2007 A preliminary version has appeared in Proc INOC, (2005), 455–461.

On computing longest paths in small graph classes

Abstract: The longest path problem is the one that finds a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. Among those, for trees, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for some tree-like graph classes by this approach. We next propose two new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on these classes.

Optimization and evaluation of shortest path queries

Abstract: We investigate the problem of how to evaluate efficiently a collection of shortest path queries on massive graphs that are too big to fit in the main memory. To evaluate a shortest path query efficiently, we introduce two pruning algorithms. These algorithms differ on the extent of materialization of shortest path cost and on how the search space is pruned. By grouping shortest path queries properly, batch processing improves the performance of shortest path query evaluation. Extensive study is also done on fragment sizes, cache sizes and query types that we show that affect the performance of a disk-based shortest path algorithm. The performance and scalability of proposed techniques are evaluated with large road systems in the Eastern United States. To demonstrate that the proposed disk-based algorithms are viable, we show that their search times are significant better than that of main-memory Dijkstra's algorithm.

On Parameterized Path and Chordless Path Problems

Abstract: We study the parameterized complexity of various path (and cycle) problems, the parameter being the length of the path. For example, we show that the problem of the existence of a maximal path of length k in a graph G is fixed-parameter tractable, while its counting version is #W[1]- complete. The corresponding problems for chordless (or induced) paths are W[2]-complete and #W[2]-complete respectively. With the tools developed in this paper we derive the NP-completeness of a related classical problem, thereby solving a problem due to Hedetniemi.

A hierarchical on-line path planning scheme using wavelets

Abstract: We present an algorithm for solving the shortest (collision-free) path planning problem for an agent (e.g., wheeled vehicle, UAV) operating in a partially known environment. The agent has detailed knowledge of the environment and the obstacles only in the vicinity of its current position. Far away obstacles or the final destination are only partially known and may even change dynamically at each instant of time. We obtain an approximation of the environment at different levels of fidelity using a wavelet approximation scheme. This allows the construction of a directed weighted graph of the obstacle-free space in a computationally efficient manner. In addition, the dimension of the graph can be adapted to the on-board computational resources. By searching this graph we find the desired shortest path to the final destination using Dijkstra's algorithm, provided that such a path exists. Simulations are presented to test the efficiency of the algorithm using non trivial scenarios.

All-pairs bottleneck paths for general graphs in truly sub-cubic time

Abstract: In the all-pairs bottleneck paths (APBP) problem (a.k.a. all-pairs maximum capacity paths), one is given a directed graph with real non-negative capacities on its edges and is asked to determine, for all pairs of vertices s and t, the capacity of a single path for which a maximum amount of flow can be routed from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and all-pairs shortest paths.We present the first truly sub-cubic algorithm for APBP in general dense graphs. In particular, we give a procedure for computing the (max, min)-product of two arbitrary matrices over R ∪ (∞,-∞) in O(n2+Ω/3) ≤ O(n2.792) time, where n is the number of vertices and Ω is the exponent for matrix multiplication over rings. Using this procedure, an explicit maximum bottleneck path for any pair of nodes can be extracted in time linear in the length of the path.

On the difficulty of some shortest path problems

Abstract: We prove superlinear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with non-negative edge weights, and a shortest path P = le1, e2, …, epr between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges ei. We show that the replacement paths problem requires Ω(m √n) time in the worst case whenever m = O(n √n). Our construction also implies a similar lower bound on the k shortest simple paths problem for a broad class of algorithms that includes all known algorithms for the problem. To put our lower bound in perspective, we note that both these problems (replacement paths and k shortest simple paths) can be solved in near-linear time for undirected graphs.

Transit functions on graphs (and posets)

Abstract: textThe notion of transit function is introduced to present a unifying approach for results and ideas on intervals, convexities and betweenness in graphs and posets. Prime examples of such transit functions are the interval function I and the induced path function J of a connected graph. Another transit function is the all-paths function. New transit functions are introduced, such as the cutvertex transit function and the longest path function. The main idea of transit functions is that of ‘transferring’ problems and ideas of one transit function to the other. For instance, a result on the interval function I might suggest similar problems for the induced path function J. Examples are given of how fruitful this transfer can be. A list of Prototype Problems and Questions for this transferring process is given, which suggests many new questions and open problems.

Efficiently determining a locally exact shortest path on polyhedral surfaces

Abstract: In this paper, we present an efficient visibility-based algorithm for determining a locally exact shortest path (LESP) from a source point to a destination point on a (triangulated) polyhedral surface. Our algorithm, of a finitely-iterative scheme, evolves an initial approximately shortest path into a LESP. During each iteration, we first compute the exact shortest path restricted on the current face sequence according to Fermat's principle which affirms that light always follows the shortest optical path, and then optimize the face sequence where the path is not locally shortest on the polyhedral surface. Since the series of paths we obtained are monotonic decreasing in length, the algorithm gives a LESP which is shorter than the initial path, at conclusion. For comparison, we use various methods to provide an initial path. One of the methods is Dijkstra's algorithm, and the others are the Fast Marching Method (FMM) and its improved version. Our intention for improvement is to overcome the limitation of acute triangulations in the original version. To achieve this goal, we classify all the edges into seven types according to different wavefront propagation manners, and dynamically determine the type of each edge for controlling the subsequent wavefront expansion. Furthermore, we give two approaches for backtracing the approximately shortest paths directed at the improved FMM. One exploits the known propagation manners of the edges as well as the Euler's method. This is another contribution in this paper.

DAG based library-free technology mapping

Abstract: This paper proposes a library-free technology mapping algorithm to reduce delay in combinational circuits. The algorithm reduces the overall number of series transistors through the longest path, considering that each cell network has to obey to a maximum admitted chain. The number of series transistors is computed in a Boolean way, reducing the structural bias. The mapping algorithm is performed on a Directed Acyclic Graph (DAG) description of the circuit. Preliminary results for delay were obtained through SPICE simulations. When compared to the SIS technology mapping, the proposed method shows significant delay reductions, considering circuits mapped with different libraries.

On shortest path representation

Abstract: Lately, it has been proposed to use shortest path first routing to implement Traffic Engineering in IP networks. The idea is to set the link weights so that the shortest paths, and the traffic thereof, follow the paths designated by the operator. Clearly, only certain shortest path representable path sets can be used in this setting, that is, paths which become shortest paths over some appropriately chosen positive, integer-valued link weights. Our main objective in this paper is to distill and unify the theory of shortest path representability under the umbrella of a novel flow-theoretic framework. In the first part of the paper, we introduce our framework and state a descriptive necessary and sufficient condition to characterize shortest path representable paths. Unfortunately, traditional methods to calculate the corresponding link weights usually produce a bunch of superfluous shortest paths, often leading to congestion along the unconsidered paths. Thus, the second part of the paper is devoted to reducing the number of paths in a representation to the bare minimum. To the best of our knowledge, this is the first time that an algorithm is proposed, which is not only able to find a minimal representation in polynomial time, but also assures link weight integrality. Moreover, we give a necessary and sufficient condition to the existence of a one-to-one mapping between a path set and its shortest path representation. However, as revealed by our simulation studies, this condition seems overly restrictive and instead, minimal representations prove much more beneficial.

Cost-based Filtering for Shorter Path Constraints

Abstract: Many real world problems, e.g. personnel scheduling and transportation planning, can be modeled naturally as Constrained Shortest Path Problems (CSPPs), i.e., as Shortest Path Problems with additional constraints. A well studied problem in this class is the Resource Constrained Shortest Path Problem. Reduction techniques are vital ingredients of solvers for the CSPP, that is frequently NP-hard, depending on the nature of the additional constraints. Viewed as heuristics, these techniques have not been studied theoretically with respect to their efficiency, i.e., with respect to the relation of filtering power and running time. Using the concepts of Constraint Programming, we provide a theoretical study of cost-based filtering for shorter path constraints on acyclic, on undirected, and on directed graphs that do not contain negative cycles. We then show empirically how reasoning about path-substructures in combination with CP-based Lagrangian relaxation can help to improve significantly over previously developed problem-tailored filtering algorithms for the resource constrained shortest path problem and investigate the impact of required-edge detection, undirected versus directed filtering, and the choice of the algorithm optimizing the Lagrangian dual.

The On-Line Shortest Path Problem Under Partial Monitoring

Abstract: The on-line shortest path problem is considered under various models of partial monitoring. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a path between two distinguished vertices such that the loss of the chosen path (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit problem, after choosing a path, the decision maker learns only the weights of those edges that belong to the chosen path. For this problem, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best path, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1= p n and depends only polynomially on the number of edges of the graph. The algorithm can be implemented with complexity that is linear in the number of rounds n (i.e., the average complexity per round is constant) and in the number of edges. An extension to the so-called label efficient setting is also given, in which the decision maker is informed about the weights of the edges corresponding to the chosen path at a total of m n time instances. Another extension is shown where the decision maker competes against a time-varying path, a generalization of the problem of tracking the best expert. A version of the multi-armed bandit setting for shortest path is also discussed where the decision maker learns only the total weight of the chosen path but not the weights of the individual edges on the path. Applications to routing in packet switched networks along with simulation results are also presented.

Logspace algorithms for computing shortest and longest paths in series-parallel graphs

Abstract: For many types of graphs, including directed acyclic graphs, undirected graphs, tournament graphs, and graphs with bounded independence number, the shortest path problem is NL-complete. The longest path problem is even NP-complete for many types of graphs, including undirected K5-minor-free graphs and planar graphs. In the present paper we present logspace algorithms for computing shortest and longest paths in series-parallel graphs where the edges can be directed arbitrarily. The class of series-parallel graphs that we study can be characterized alternatively as the class of K4-minor-free graphs and also as the class of graphs of tree-width 2. It is well-known that for graphs of bounded tree-width many intractable problems can be solved efficiently, but previous work was focused on finding algorithms with low parallel or sequential time complexity. In contrast, our results concern the space complexity of shortest and longest path problems. In particular, our results imply that for directed graphs of tree-width 2 these problems are L-complete.

Inapproximability results for the inverse shortest paths problem with integer lengths and unique shortest paths

Abstract: We study the complexity of two inverse shortest paths (ISP) problems with integer arc lengths and the requirement for uniquely determined shortest paths. Given a collection of paths in a directed graph D = (V, A), the task is to find positive integer arc lengths such that the given paths are uniquely determined shortest paths between their respective terminals. In the first problem we seek for arc lengths that minimize the length of the longest of the prescribed paths. In the second problem, the length of the longest arc is to be minimized. We show that it is 5©5«-hard to approximate the minimal longest path length within a factor less than 8-7 or the minimal longest arc length within a factor less than 9-8. This answers the (previously) open question whether these problems are 5©5«-hard or not. We also present a simple algorithm that achieves an 5a(|V|)-approximation guarantee for both variants. Both ISP problems arise in the planning of telecommunication networks with shortest path routing protocols. Our results imply that it is 5©5«-hard to decide whether a given path set can be realized with a real shortest path routing protocol such as OSPF, IS-IS, or RIP. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 2936 2007