Showing papers on "Longest path problem published in 2015"

An exact algorithm for the mean–standard deviation shortest path problem

Abstract: This paper studies the reliable path problem in the form of minimizing the sum of mean and standard deviation of path travel time. For the case of independent link travel times, we show that the problem can be solved exactly by repeatedly solving a subproblem minimizing the sum of mean and variance of path travel time. The latter is an additive shortest path problem, and can be solved using a standard labeling algorithm. While these subproblems are similar in form to those obtained from Lagrangian relaxation, this formulation admits proof of finite convergence to the optimal solution. An iterative labeling algorithm is developed that solves the non-additive reliable path problem from a single origin to all destinations. Moreover, a labeling technique is employed to further reduce the computational time of the proposed algorithm by partially updating the network in each iteration. As an alternative, a bisection-type search algorithm is developed that solves the problem for the single-origin and single-destination case. Numerical tests are presented, indicating that the proposed algorithm outperform others recently proposed in the literature: unlike Lagrangian relaxation, two of the proposed algorithms find solutions exactly, and the computation time is an order of magnitude faster than outer approximation methods.

Algorithmic Applications of Baur-Strassen’s Theorem: Shortest Cycles, Diameter, and Matchings

Abstract: Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this article, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen’s theorem and of Strojohann’s determinant algorithm. It allows us to give new and simple solutions to the following problems: Finding Shortest Cycles. We give a simple O(Wnω) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with nonnegative weights), this matches the time bounds obtained in 2011 by Roditty and Williams. On the other hand, no algorithm working in O(Wn ω) time was previously known for undirected graphs with negative weights. Furthermore, our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. Computing Diameter and Radius. We give a simple O(Wnω) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge, no algorithm with this running time was known for undirected graphs with negative weights. Finding Minimum-Weight Perfect Matchings. We present an O(Wnω) time algorithm for finding minimum-weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski [2009] who presented such an algorithm but only in the case of bipartite graphs. These three problems that are solved in the full generality demonstrate the utility of this framework. Hence, we believe that it can find applications for solving larger spectra of related problems. As an illustrative example, we apply it to the problem of computing a set of vertices that lie on cycles of length at most t, for some given t. We give a simple O(Wnω) time algorithm for this problem that improves over the O(Wnωt) time algorithm given by Yuster in 2011. Besides giving this flexible framework, the other main contribution of this article is the development of a novel combinatorial interpretation of the dual solution for the minimum-weight perfect matching problem. Despite the long history of the matching problem, such a combinatorial interpretation was not known previously. This result sheds a new light on the problem, as there exist many structural theorems about unweighted matchings, but almost no results that could cope with the weighted case.

On the Quadratic Shortest Path Problem

Abstract: Finding the shortest path in a directed graph is one of the most important combinatorial optimization problems, having applications in a wide range of fields. In its basic version, however, the problem fails to represent situations in which the value of the objective function is determined not only by the choice of each single arc, but also by the combined presence of pairs of arcs in the solution. In this paper we model these situations as a Quadratic Shortest Path Problem, which calls for the minimization of a quadratic objective function subject to shortest-path constraints. We prove strong NP-hardness of the problem and analyze polynomially solvable special cases, obtained by restricting the distance of arc pairs in the graph that appear jointly in a quadratic monomial of the objective function. Based on this special case and problem structure, we devise fast lower bounding procedures for the general problem and show computationally that they clearly outperform other approaches proposed in the literature in terms of their strength.

How Well Can Graphs Represent Wireless Interference

Abstract: Efficient use of a wireless network requires that transmissions be grouped into feasible sets, where feasibility means that each transmission can be successfully decoded in spite of the interference caused by simultaneous transmissions. Feasibility is most closely modeled by a signal-to-interference-plus-noise (SINR) formula, which unfortunately is conceptually complicated, being an asymmetric, cumulative, many-to-one relationship. We re-examine how well graphs can capture wireless receptions as encoded in SINR relationships, placing them in a framework in order to understand the limits of such modelling. We seek for each wireless instance a pair of graphs that provide upper and lower bounds on the feasibility relation, while aiming to minimize the gap between the two graphs. The cost of a graph formulation is the worst gap over all instances, and the price of (graph) abstraction is the smallest cost of a graph formulation. We propose a family of conflict graphs that is parameterized by a non-decreasing sub-linear function, and show that with a judicious choice of functions, the graphs can capture feasibility with a cost of O(log* Δ), where Δ is the ratio between the longest and the shortest link length. This holds on the plane and more generally in doubling metrics. We use this to give greatly improved O(log* Δ)-approximation for fundamental link scheduling problems with arbitrary power control. We also explore the limits of graph representations and find that our upper bound is tight: the price of graph abstraction is Ω(log* Δ). In addition, we give strong impossibility results for general metrics, and for approximations in terms of the number of links.

Faster Algorithms for Algebraic Path Properties in Recursive State Machines with Constant Treewidth

Abstract: Interprocedural analysis is at the heart of numerous applications in programming languages, such as alias analysis, constant propagation, etc. Recursive state machines (RSMs) are standard models for interprocedural analysis. We consider a general framework with RSMs where the transitions are labeled from a semiring, and path properties are algebraic with semiring operations. RSMs with algebraic path properties can model interprocedural dataflow analysis problems, the shortest path problem, the most probable path problem, etc. The traditional algorithms for interprocedural analysis focus on path properties where the starting point is fixed as the entry point of a specific method. In this work, we consider possible multiple queries as required in many applications such as in alias analysis. The study of multiple queries allows us to bring in a very important algorithmic distinction between the resource usage of the one-time preprocessing vs for each individual query. The second aspect that we consider is that the control flow graphs for most programs have constant treewidth. Our main contributions are simple and implementable algorithms that support multiple queries for algebraic path properties for RSMs that have constant treewidth. Our theoretical results show that our algorithms have small additional one-time preprocessing, but can answer subsequent queries significantly faster as compared to the current best-known solutions for several important problems, such as interprocedural reachability and shortest path. We provide a prototype implementation for interprocedural reachability and intraprocedural shortest path that gives a significant speed-up on several benchmarks.

Robust Optimization Strategy for the Shortest Path Problem under Uncertain Link Travel Cost Distribution

Abstract: This article showed how numerical experiments conducted on small to large networks compare the robust optimization-based strategy to the classical deterministic shortest path in terms of the uncertainty. A robust optimization approach for the shortest path problem where travel cost is uncertain and exact information on the distribution function is unavailable is developed. The article showed that under such conditions the robust shortest path problem can be formulated as a binary nonlinear integer program, which can then be reformulated as a mixed integer conic quadratic program. This article presented an outer approximation algorithm as a solution algorithm, which is shown to be highly efficient for this class of programs.

The safest path via safe zones

Abstract: We define and study Euclidean and spatial network variants of a new path finding problem: given a set of safe zones, find paths that minimize the distance traveled outside the safe zones. In this problem, the entire space with the exception of the safe zones is unsafe, but passable, and it differs from problems that involve unsafe regions to be strictly avoided. As a result, existing algorithms are not effective solutions to the new problem.

Multi-agent path finding on strongly biconnected digraphs

Abstract: Much of the literature on multi-agent path finding focuses on undirected graphs, where motion is permitted in both directions along a graph edge. Despite this, travelling on directed graphs is relevant in navigation domains, such as pathfinding in games, and asymmetric communication networks. We consider multi-agent path finding on strongly biconnected directed graphs. We show that all instances with at least two unoccupied positions can be solved or proven unsolvable. We present a polynomial-time algorithm for this class of problems, and analyze its complexity. Our work may be the first formal study of multi-agent path finding on directed graphs.

Snakes, coils, and single-track circuit codes with spread $$k$$k

Abstract: The snake-in-the-box problem is concerned with finding a longest induced path in a hypercube $$Q_n$$Qn. Similarly, the coil-in-the-box problem is concerned with finding a longest induced cycle in $$Q_n$$Qn. We consider a generalization of these problems that considers paths and cycles where each pair of vertices at distance at least $$k$$k in the path or cycle are also at distance at least $$k$$k in $$Q_n$$Qn. We call these paths $$k$$k-snakes and the cycles $$k$$k-coils. The $$k$$k-coils have also been called circuit codes. By optimizing an exhaustive search algorithm, we find 13 new longest $$k$$k-coils, 21 new longest $$k$$k-snakes and verify that some of them are optimal. By optimizing an algorithm by Paterson and Tuliani to find single-track circuit codes, we additionally find another 8 new longest $$k$$k-coils. Using these $$k$$k-coils with some basic backtracking, we find 18 new longest $$k$$k-snakes.

Generating random graphs in biased Maker-Breaker games

Abstract: We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for b=on, Maker can build a pancyclic graph that is, a graph that contains cycles of every possible length while playing a 1:b game on EKn. As another application, we show that for b=i¾?n/lnn, playing a 1:b game on EKn, Maker can build a graph which contains copies of all spanning trees having maximum degree Δ=O1 with a bare path of linear length a bare path in a tree T is a path with all interior vertices of degree exactly two in T. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 47, 615-634, 2015

Finding Paths in Grids with Forbidden Transitions

Abstract: A transition in a graph is a pair of adjacent edges. Given a graph $$G=V,E$$, a set of forbidden transitions $$\mathcal{F}\subseteq E\times E$$ and two vertices $$s,t \in V$$, we study the problem of finding a path from s to t which uses none of the forbidden transitions of $$\mathcal F$$. This means that it is forbidden for the path to consecutively use two edges forming a pair ini¾?$$\mathcal F$$. The study of this problem is motivated by routing in road networks in which forbidden transitions are associated to prohibited turns as well as routing in optical networks with asymmetric nodes, which are nodes where a signal on an ingress port can only reach a subset of egress ports. If the path is not required to be elementary, the problem can be solved in polynomial time. On the other side, if the path has to be elementary, the problem is known to be NP-complete in general graphs [Szeider 2003]. In this paper, we study the problem of finding an elementary path avoiding forbidden transitions in planar graphs. We prove that the problem is NP-complete in planar graphs and particularly in grids. In addition, we show that the problem can be solved in polynomial time in graphs with bounded treewidth. More precisely, we show that there is an algorithm which solves the problem in time $$Ok\varDelta ^23 k \varDelta ^{2k} n$$ in n-node graphs with treewidth at most k and maximum degree $$\varDelta $$.

On 4-Map Graphs and 1-Planar Graphs and their Recognition Problem

Abstract: We establish a one-to-one correspondence between 1-planar graphs and general and hole-free 4-map graphs and show that 1-planar graphs can be recognized in polynomial time if they are crossing-augmented, fully triangulated, and maximal 1-planar, respectively, with a polynomial of degree 120, 3, and 5, respectively.

Entropy weighted average method for the determination of a single representative path flow solution for the static user equilibrium traffic assignment problem

Abstract: The formulation of the static user equilibrium traffic assignment problem (UETAP) under some simplifying assumptions has a unique solution in terms of link flows but not in terms of path flows. Large variations are possible in the path flows obtained using different UETAP solution algorithms. Many transportation planning and management applications entail the need for path flows. This raises the issue of generating a meaningful path flow solution in practice. Past studies have sought to determine a single path flow solution using the maximum entropy concept. This study proposes an alternate approach to determine a single path flow solution that represents the entropy weighted average of the UETAP path flow solution space. It has the minimum expected Euclidean distance from all other path flow solution vectors of the UETAP. The mathematical model of the proposed entropy weighted average method is derived and its solution stability is proved. The model is easy to interpret and generalizes the proportionality condition of Bar-Gera and Boyce (1999). Results of numerical experiments using networks of different sizes suggest that the path flow solutions for the UETAP using the proposed method are about identical to those obtained using the maximum entropy approach. The entropy weighted average method requires low computational effort and is easier to implement, and can therefore serve as a potential alternative to the maximum entropy approach in practice.

The edge-disjoint path problem on random graphs by message-passing

Abstract: We present a message-passing algorithm to solve a series of edge-disjoint path problems on graphs based on the zero-temperature cavity equations. Edge-disjoint paths problems are important in the general context of routing, that can be defined by incorporating under a unique framework both traffic optimization and total path length minimization. The computation of the cavity equations can be performed efficiently by exploiting a mapping of a generalized edge-disjoint path problem on a star graph onto a weighted maximum matching problem. We perform extensive numerical simulations on random graphs of various types to test the performance both in terms of path length minimization and maximization of the number of accommodated paths. In addition, we test the performance on benchmark instances on various graphs by comparison with state-of-the-art algorithms and results found in the literature. Our message-passing algorithm always outperforms the others in terms of the number of accommodated paths when considering non trivial instances (otherwise it gives the same trivial results). Remarkably, the largest improvement in performance with respect to the other methods employed is found in the case of benchmarks with meshes, where the validity hypothesis behind message-passing is expected to worsen. In these cases, even though the exact message-passing equations do not converge, by introducing a reinforcement parameter to force convergence towards a sub optimal solution, we were able to always outperform the other algorithms with a peak of 27% performance improvement in terms of accommodated paths. On random graphs, we numerically observe two separated regimes: one in which all paths can be accommodated and one in which this is not possible. We also investigate the behavior of both the number of paths to be accommodated and their minimum total length.

Compressing optimal paths with run length encoding

Abstract: We introduce a novel approach to Compressed Path Databases, space efficient oracles used to very quickly identify the first edge on a shortest path. Our algorithm achieves query running times on the 100 nanosecond scale, being significantly faster than state-of-the-art first-move oracles from the literature. Space consumption is competitive, due to a compression approach that rearranges rows and columns in a first-move matrix and then performs run length encoding (RLE) on the contents of the matrix. One variant of our implemented system was, by a convincing margin, the fastest entry in the 2014 Grid-Based Path Planning Competition. We give a first tractability analysis for the compression scheme used by our algorithm. We study the complexity of computing a database of minimum size for general directed and undirected graphs. We find that in both cases the problem is NP-complete. We also show that, for graphs which can be decomposed along articulation points, the problem can be decomposed into independent parts, with a corresponding reduction in its level of difficulty. In particular, this leads to simple and tractable algorithms with linear running time which yield optimal compression results for trees.

Linear path skylines in multicriteria networks

Abstract: In many graph applications, computing cost-optimal paths between two locations is an important task for routing and distance computation. Depending on the network multiple cost criteria might be of interest. Examples are travel time, energy consumption and toll fees in road networks. Path skyline queries compute the set of pareto optimal paths between two given locations. However, the number of skyline paths increases exponentially with the distance between the locations and the number of cost criteria. Thus, the result set might be too big to be of any use. In this paper, we introduce multicriteria linear path skyline queries. A linear path skyline is the subset of the conventional path skyline where the paths are optimal under a linear combination of their cost values. We argue that cost vectors being optimal with respect to a weighted sum are intuitive to understand and therefore, more interesting in many cases. We show that linear path skylines are convex hulls of an augmented solution space and propose an algorithm which utilizes this observation to efficiently compute the complete linear path skyline. To further control the size of the result set, we introduce an approximate version of our algorithm guaranteeing a certain level of optimality for each possible weighting. In our experimental evaluation, we show that our approach computes linear path skylines significantly faster than previous approaches, including those computing the complete path skyline.

Automatically computing path complexity of programs

Abstract: Recent automated software testing techniques concentrate on achieving path coverage. We present a complexity measure that provides an upper bound for the number of paths in a program, and hence, can be used for assessing the difficulty of achieving path coverage for a given method. We define the path complexity of a program as a function that takes a depth bound as input and returns the number of paths in the control flow graph that are within that bound. We show how to automatically compute the path complexity function in closed form, and the asymptotic path complexity which identifies the dominant term in the path complexity function. Our results demonstrate that path complexity can be computed efficiently, and it is a better complexity measure for path coverage compared to cyclomatic complexity and NPATH complexity.

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

Abstract: We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0

On the Minimum Eccentricity Shortest Path Problem

Abstract: In this paper, we introduce and investigate the Minimum Eccentricity Shortest Path (MESP) problem in unweighted graphs. It asks for a given graph to find a shortest path with minimum eccentricity. We demonstrate that: a minimum eccentricity shortest path plays a crucial role in obtaining the best to date approximation algorithm for a minimum distortion embedding of a graph into the line; the MESP-problem is NP-hard on general graphs; a 2-approximation, a 3-approximation, and an 8-approximation for the MESP-problem can be computed in \(\mathcal {O}(n^3)\) time, in \(\mathcal {O}(nm)\) time, and in linear time, respectively; a shortest path of minimum eccentricity k in general graphs can be computed in \(\mathcal {O}(n^{2k+2}m)\) time; the MESP-problem can be solved in linear time for trees.

Three-coloring graphs with no induced seven-vertex path II : using a triangle.

Abstract: In this paper, we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. In previous work, we gave a polynomial time algorithm for three-coloring triangle-free graphs with no induced seven-vertex path. Combined, our work shows that three-coloring a graph with no induced seven-vertex path can be done in polynomial time.

Protected shortest path visiting specified nodes

Abstract: In this paper heuristics are proposed for finding the shortest loopless path, from a source node to a target node, that visits a given set of nodes in a directed graph, such that it can be protected using a node-disjoint path. This type of problem may arise due to network management constraints. The problem of calculating the shortest path that visits a given set of nodes is at least as difficult as the traveling salesman problem, and it has not received much attention. Nevertheless an efficient integer linear programming (ILP) formulation has been recently proposed for this problem. Here, the ILP formulation is adapted to include the constraint that the obtained path will be able to be protected by a node-disjoint path. Computational experiments show that this approach, namely in large networks, may fail to obtain a solution in a reasonable amount of time. Therefore three versions of a heuristic are proposed, for which extensive computational results show that they are able to find a solution in most cases, and that the calculated solutions present an acceptable relative error regarding the cost of the optimal active path. Further the CPU time required by the heuristics is significantly smaller than the required by the used ILP solver.

Safe Path Planning Using Cell Decomposition Approximation

Abstract: Motion planning is an essential part in robotics domain; it is responsible for guiding the robot motion toward the goal. It generates a path from one location to another one, while avoiding the obstacles in the way. The planning modules could be configured to check the optimality, completeness, power saving, shortness of path, minimal number of turn, or the turn sharpness, etc., in addition to path safety. In this paper the cell decomposition approximation planar is used to find a safe path; the quad-tree approximation algorithm divides the workspace into manageable free areas, and builds a graph of adjacency between them. New methods are proposed to keep the robot far away from the obstacles boundaries by a minimum safe distance. These methods manipulate the weights of adjacency graph's edges. They utilize and reflect the size of free cells when planning a path. These approaches give a lower weight to the connection between big free cells, and a higher weight to the connections between the smaller cells. The planner after that searches for the lowest cost path based on these weights. The safe path in this work is the path which keeps the robot far away from obstacles by specified minimum safety distance and it bias the robot's motion to follow the bigger areas in the workspace. The shortest path is not considered. However a tradeoff between the real path cost and the safe path cost is considered when choosing the weight values.

Entropy Weighted Average Method for the Determination of Single Representative Path Flow Solution for the Static User Equilibrium Traffic Assignment Problem

Abstract: The formulation for the static user equilibrium traffic assignment problem (UETAP) under some simplifying assumptions has a unique solution in terms of link flows but not in terms of path flows. The associated solution algorithms can illustrate large variations in terms of path flows. Many transportation planning and management applications entail the need for path flows. This raises the issue of generating a meaningful path flow solution for practice. Past studies have sought to determine single path flow solution from the set multiple solutions using the maximum entropy concept. This study proposes an alternate approach to determine single path flow solution that represents the entropy weighted average of the UETAP path flow solution space. It has the minimum expected Euclidean distance from all other path flow solution vectors of the UETAP. The mathematical model of the proposed entropy weighted average method is derived. The model is easy to interpret and generalizes the proportionality condition proposed by Bar-Gera and Boyce in past. Results of numerical experiments using networks of different sizes suggest that the path flow solutions for the UETAP using the proposed method are about identical to those obtained using the maximum entropy approach. The entropy weighted average method requires low computational effort and is easier to implement, and can therefore serve as a potential alternative to the maximum entropy approach for practice.

A combination of flow shop scheduling and the shortest path problem

Abstract: This paper studies a combinatorial optimization problem which is obtained by combining the flow shop scheduling problem and the shortest path problem. The objective of the obtained problem is to select a subset of jobs that constitutes a feasible solution to the shortest path problem, and to execute the selected jobs on the flow shop machines to minimize the makespan. We argue that this problem is NP-hard even if the number of machines is two, and is NP-hard in the strong sense for the general case. We propose an intuitive approximation algorithm for the case where the number of machines is an input, and an improved approximation algorithm for fixed number of machines.

Maximum Weight Independent Sets for ($P_7$,Triangle)-Free Graphs in Polynomial Time

Abstract: The Maximum Weight Independent Set (MWIS) problem on finite undirected graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum weight sum. MWIS is one of the most investigated and most important algorithmic graph problems; it is well known to be NP-complete, and it remains NP-complete even under various strong restrictions such as for triangle-free graphs. Its complexity was an open problem for $P_k$-free graphs, $k \ge 5$. Recently, Lokshtanov, Vatshelle, and Villanger proved that MWIS can be solved in polynomial time for $P_5$-free graphs, and Lokshtanov, Pilipczuk, and van Leeuwen proved that MWIS can be solved in quasi-polynomial time for $P_6$-free graphs. It still remains an open problem whether MWIS can be solved in polynomial time for $P_k$-free graphs, $k \geq 6$ or in quasi-polynomial time for $P_k$-free graphs, $k \geq 7$. Some characterizations of $P_k$-free graphs and some progress are known in the literature but so far did not solve the problem. In this paper, we show that MWIS can be solved in polynomial time for ($P_7$,triangle)-free graphs. This extends the corresponding result for ($P_6$,triangle)-free graphs and may provide some progress in the study of MWIS for $P_7$-free graphs.