Showing papers on "Longest path problem published in 2011"
TL;DR: An algorithmic approach is developed exploiting the graph-theoretic properties of a k-plex that is effective in solving the problem to optimality on very large, sparse graphs such as the power law graphs frequently encountered in the applications of interest.
Abstract: This paper introduces and studies the maximum k-plex problem, which arises in social network analysis and has wider applicability in several important areas employing graph-based data mining. After establishing NP-completeness of the decision version of the problem on arbitrary graphs, an integer programming formulation is presented, followed by a polyhedral study to identify combinatorial valid inequalities and facets. A branch-and-cut algorithm is implemented and tested on proposed benchmark instances. An algorithmic approach is developed exploiting the graph-theoretic properties of a k-plex that is effective in solving the problem to optimality on very large, sparse graphs such as the power law graphs frequently encountered in the applications of interest.
300 citations
Book•
08 Sep 2011
TL;DR: The main result of this paper involves a favorable special case of the 3-D shortest path problem, namely that of finding the shortest path between two points along the surface of a convex polyhedron.
Abstract: We consider the problem of computing the shortest path between two points in two- or three-dimensional space bounded by polyhedral surfaces. In the 2-D case the problem is easily solved in time O(n2 log n).In the general 3-D case the problem is quite hard to solve, and is not even discrete; we present a doubly-exponential procedure for solving the discrete subproblem of determining the sequence of boundary edges through which the shortest path passes. The main result of this paper involves a favorable special case of the 3-D shortest path problem, namely that of finding the shortest path between two points along the surface of a convex polyhedron. We analyze this problem and solve it in time O(n3 log n).
193 citations
TL;DR: It is pointed out that there exists an equivalence relation between the @a-shortest path in an uncertain network and the shortest path in a corresponding deterministic network, which leads to an effective algorithm to find the@a- shortest path and the most shortest path.
Abstract: Uncertainty theory provides a new tool to deal with the shortest path problem with nondeterministic arc lengths. With help from the operational law of uncertainty theory, this paper gives the uncertainty distribution of the shortest path length. Also, it investigates solutions to the @a-shortest path and the most shortest path in an uncertain network. It points out that there exists an equivalence relation between the @a-shortest path in an uncertain network and the shortest path in a corresponding deterministic network, which leads to an effective algorithm to find the @a-shortest path and the most shortest path. Roughly speaking, this algorithm can be broken down into two parts: constructing a deterministic network and then invoking the Dijkstra algorithm.
167 citations
10 Apr 2011
TL;DR: This paper transforms the problem of maximizing the lifetime of data aggregation trees, which are limited to shortest path trees, into a general version of semi-matching problem, and shows that the problem can be solved by min-cost max-flow approach in polynomial time.
Abstract: In many applications of wireless sensor networks, a sensor node senses the environment to get data and delivers them to the sink via a single hop or multi-hop path. Many systems use a tree rooted at the sink as the underlying routing structure. Since the sensor node is energy constrained, how to construct a good tree to prolong the lifetime of the network is an important problem. We consider this problem under the scenario where nodes have different initial energy, and they can do in-network aggregation. In previous works, it has been proved that finding a maximum lifetime tree from all feasible spanning trees is NP-complete. Since delay is also an important element in time-critical applications, and shortest path trees intuitively have short delay, it is imperative to find a shortest path tree with long lifetime. This paper studies the problem of maximizing the lifetime of data aggregation trees, which are limited to shortest path trees. We find that when it is restricted to shortest path trees, the original problem is in P. We transform the problem into a general version of semi-matching problem, and show that the problem can be solved by min-cost max-flow approach in polynomial time. Also we design a distributed solution. Simulation results show that our approach greatly improves the lifetime of the network and is more competitive when it is applied in a dense network.
115 citations
10 Apr 2011
TL;DR: It is shown that, using O(k log(n)) path measurements, it is able to recover any k-sparse link vector (with no more than k nonzero elements), even though the measurements have to follow the graph path constraints.
Abstract: In this paper, motivated by network inference and tomography applications, we study the problem of compressive sensing for sparse signal vectors over graphs. In particular, we are interested in recovering sparse vectors representing the properties of the edges from a graph. Unlike existing compressive sensing results, the collective additive measurements we are allowed to take must follow connected paths over the underlying graph. For a sufficiently connected graph with n nodes, it is shown that, using O(k log(n)) path measurements, we are able to recover any k-sparse link vector (with no more than k nonzero elements), even though the measurements have to follow the graph path constraints. We mainly show that the computationally efficient l 1 minimization can provide theoretical guarantees for inferring such k-sparse vectors with O(k log(n)) path measurements from the graph.
114 citations
TL;DR: It is proved that the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints is strongly NP -hard, even if every connected component of the conflict graph is a path of length two.
89 citations
24 Oct 2011
TL;DR: This paper presents two improvements to existing landmark-based shortest path estimation methods that relate to the use of shortest-path trees (SPTs) and a new landmark selection strategy that seeks to maximize the coverage of all shortest paths by the selected landmarks.
Abstract: Computing the shortest path between a pair of vertices in a graph is a fundamental primitive in graph algorithmics. Classical exact methods for this problem do not scale up to contemporary, rapidly evolving social networks with hundreds of millions of users and billions of connections. A number of approximate methods have been proposed, including several landmark-based methods that have been shown to scale up to very large graphs with acceptable accuracy. This paper presents two improvements to existing landmark-based shortest path estimation methods. The first improvement relates to the use of shortest-path trees (SPTs). Together with appropriate short-cutting heuristics, the use of SPTs allows to achieve higher accuracy with acceptable time and memory overhead. Furthermore, SPTs can be maintained incrementally under edge insertions and deletions, which allows for a fully-dynamic algorithm. The second improvement is a new landmark selection strategy that seeks to maximize the coverage of all shortest paths by the selected landmarks. The improved method is evaluated on the DBLP, Orkut, Twitter and Skype social networks.
87 citations
TL;DR: A new algorithm for the efficient generation of all non-isomorphic connected cubic graphs that is more than 4 times faster than previous generators and can be efficiently restricted to cubic graphs with girth at least 4 or 5.
Abstract: We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5.
62 citations
TL;DR: This paper first models the MCPPR as a particular bicriteria path problem involving an aggregated function of the path and relay costs, as well as a weight function, and considers a variant of this problem which takes into account all three functions separately.
61 citations
TL;DR: A new suboptimal offline path search algorithm based on the A^* algorithm, which takes the threat zones in the game map into consideration and searches for the shortest path from a starting location to a destination, where the agent may suffer damage less than or equal to the specified limit.
Abstract: Pathfinding algorithms used in todays computer games consider the path length or a similar criterion as the only measure of optimality. However, these games usually involve opposing parties, whose agents can inflict damage on those of the others. Therefore, the shortest path in such games may not always be the safest one. Consequently, a new suboptimal offline path search algorithm based on the A^* algorithm was developed, which takes the threat zones in the game map into consideration. Given an upper limit as the tolerable amount of damage for an agent, this algorithm searches for the shortest path from a starting location to a destination, where the agent may suffer damage less than or equal to the specified limit. Due to its behavior, the algorithm is called Limited-Damage A^* (LDA^*). Performance of LDA^* was tested in randomly-generated maze-like grid-based environments of varying sizes, and in hand-crafted fully-observable environments, in which 8-way movement is utilized. Results obtained from LDA^* are compared with those obtained from Multiobjective A^* (MOA^*), which is a complete and optimal algorithm that yields exact (best) solutions for every case. LDA^* was found to perform much faster than MOA^*, yielding acceptable sub-optimality in path length.
55 citations
01 Dec 2011
TL;DR: Based on the observation that a variety of graph search queries can be implemented by iterative operations including selecting frontier nodes from visited nodes, making expansion from the selected frontier nodes, and merging the expanded nodes into the visited ones, the authors introduced a relational FEM framework with three corresponding operators to implement graph search tasks in the RDB context.
Abstract: With the rapid growth of large graphs, we cannot assume that graphs can still be fully loaded into memory, thus the disk-based graph operation is inevitable. In this paper, we take the shortest path discovery as an example to investigate the technique issues when leveraging existing infrastructure of relational database (RDB) in the graph data management.Based on the observation that a variety of graph search queries can be implemented by iterative operations including selecting frontier nodes from visited nodes, making expansion from the selected frontier nodes, and merging the expanded nodes into the visited ones, we introduce a relational FEM framework with three corresponding operators to implement graph search tasks in the RDB context. We show new features such as window function and merge statement introduced by recent SQL standards can not only simplify the expression but also improve the performance of the FEM framework. In addition, we propose two optimization strategies specific to shortest path discovery inside the FEM framework. First, we take a bi-directional set Dijkstra's algorithm in the path finding. The bi-directional strategy can reduce the search space, and set Dijkstra's algorithm finds the shortest path in a set-at-a-time fashion. Second, we introduce an index named SegTable to preserve the local shortest segments, and exploit SegTable to further improve the performance. The final extensive experimental results illustrate our relational approach with the optimization strategies achieves high scalability and performance.
TL;DR: It is shown that the PDTC problem can be solved in polynomial time when the number s of stacks is fixed but the size of each stack is not, and the bounded coloring (BC) problem on permutation graphs is built upon.
TL;DR: The longest path problem can be solved in polynomial time on interval graphs with a dynamic programming approach and runs in O(n4) time, where n is the number of vertices of the input graph.
Abstract: The longest path problem is the problem of finding a path of maximum length in a graph. Polynomial solutions for this problem are known only for small classes of graphs, while it is NP-hard on general graphs, as it is a generalization of the Hamiltonian path problem. Motivated by the work of Uehara and Uno (Proc. of the 15th Annual International Symp. on Algorithms and Computation (ISAAC), LNCS, vol. 3341, pp. 871–883, 2004), where they left the longest path problem open for the class of interval graphs, in this paper we show that the problem can be solved in polynomial time on interval graphs. The proposed algorithm uses a dynamic programming approach and runs in O(n 4) time, where n is the number of vertices of the input graph.
TL;DR: The TSP is considered as a problem formulation for the multi-goal path planning problem in which paths among obstacles have to be found and a simple approximation of the shortest path seems to be suitable for the SOM adaptation procedure.
TL;DR: It is found that in 11 of the 12 species, average path lengths in PPI networks are significantly longer than those in randomly rewired networks, and node eccentricity is significantly correlated with the levels of gene expression and dispensability in the yeast PPI network.
Abstract: We investigated the biological significance of path lengths in 12 protein-protein interaction (PPI) networks. We put forward three predictions, based on the idea that biological complexity influences path lengths. First, at the network level, path lengths are generally longer in PPIs than in random networks. Second, this pattern is more pronounced in more complex organisms. Third, within a PPI network, path lengths of individual proteins are biologically significant. We found that in 11 of the 12 species, average path lengths in PPI networks are significantly longer than those in randomly rewired networks. The PPI network of the malaria parasite Plasmodium falciparum, however, does not exhibit deviation from rewired networks. Furthermore, eukaryotic PPIs exhibit significantly greater deviation from randomly rewired networks than prokaryotic PPIs. Thus our study highlights the potentially meaningful variation in path lengths of PPI networks. Moreover, node eccentricity, defined as the longest path from a protein to others, is significantly correlated with the levels of gene expression and dispensability in the yeast PPI network. We conclude that biological complexity influences both global and local properties of path lengths in PPI networks. Investigating variation of path lengths may provide new tools to analyze the evolution of functional modules in biological systems.
TL;DR: This work provides sufficient conditions for a node and an arc to be always or never in an optimal solution of the Minimax regret Single-Pair Shortest Path Problem (MSP-SPP), and exploits such results to develop pegging tests useful to reduce the overall running time necessary to exactly solve both problems.
TL;DR: After formulating the shortest path problem as a minimum-time problem, a family of extremals, which is sufficient for optimality, is characterized, and subsequently the complete analytic solution of the optimal synthesis problem is presented.
Abstract: We consider a variation of the classical Markov–Dubins problem dealing with curvature-constrained, shortest paths in the plane with prescribed initial and terminal positions and tangents, when the lower and upper bounds of the curvature of the path are not necessarily equal. The motivation for this problem stems from vehicle navigation applications, when a vehicle may be biased in taking turns at a particular direction due to hardware failures or environmental conditions. After formulating the shortest path problem as a minimum-time problem, a family of extremals, which is sufficient for optimality, is characterized, and subsequently the complete analytic solution of the optimal synthesis problem is presented. In addition, the synthesis problem, when the terminal tangent is free, is also considered, leading to the characterization of the set of points that can be reached in the plane by curves satisfying asymmetric curvature constraints.
24 Aug 2011
TL;DR: The proposed approach reduces this redundancy by determining the critical time points at which an optimal path may change, and theoretical analysis and experimental results show that this approach performs better than naive approaches particularly when there are fewcritical time points.
Abstract: Given a spatio-temporal network, a source, a destination, and a start-time interval, the All-start-time Lagrangian Shortest Paths (ALSP) problem determines a path set which includes the shortest path for every start time in the given interval. ALSP is important for critical societal applications related to air travel, road travel, and other spatiotemporal networks. However, ALSP is computationally challenging due to the non-stationary ranking of the candidate paths, meaning that a candidate path which is optimal for one start time may not be optimal for others. Determining a shortest path for each start-time leads to redundant computations across consecutive start times sharing a common solution. The proposed approach reduces this redundancy by determining the critical time points at which an optimal path may change. Theoretical analysis and experimental results show that this approach performs better than naive approaches particularly when there are few critical time points.
TL;DR: This work proposes an algorithm that finds in an n -vertex grid graph G, a long path of length at least 5 6 n + 2 , and runs in quadratic time.
TL;DR: This paper investigates the depth of resolution proofs, that is to say, the length of the longest path in the proof from an input clause to the conclusion.
Abstract: This paper investigates the depth of resolution proofs, that is to say, the length of the longest path in the proof from an input clause to the conclusion. An abstract characterization of the measure is given, as well as a discussion of its relation to other measures of space complexity for resolution proofs
TL;DR: This paper presents O ( n ) -time certifying algorithms for the above two problems on interval graphs given a set of n intervals with endpoints sorted that can be authenticated in O (n ) time.
20 Nov 2011
TL;DR: A versatile method that enumerates all or a user-specified number of longest sensitisable paths in the whole circuit or through specific components, which allows the method to benefit from recent advances in SAT-solving technology, but also to avoid some of the drawbacks of previous structural approaches.
Abstract: We present a versatile method that enumerates all or a user-specified number of longest sensitisable paths in the whole circuit or through specific components. The path information can be used for design and test of circuits affected by statistical process variations. The algorithm encodes all aspects of the path search as an instance of the Boolean Satisfiability Problem (SAT), which allows the method not only to benefit from recent advances in SAT-solving technology, but also to avoid some of the drawbacks of previous structural approaches. Experimental results for academic and industrial benchmark circuits demonstrate the method's accuracy and scalability.
23 Jan 2011
TL;DR: In this paper, the authors showed that for small integer weights, the replacement paths problem in directed graphs may be easier than the related all pairs shortest paths problem for directed graphs, as the current best runtime for the latter is Ω(n25) time even if ω = 2.
Abstract: The replacement paths problem for directed graphs is to find for given nodes s and t and every edge e on the shortest path between them, the shortest path between s and t which avoids e For unweighted directed graphs on n vertices, the best known algorithm runtime was O(n25) by Roditty and Zwick For graphs with integer weights in {− M,, M}, Weimann and Yuster showed that one can use fast matrix multiplication and solve the problem in O(Mn2584) time, a runtime which would be O(Mn233) if the exponent ω of matrix multiplication is 2We improve on both of these algorithms Our new algorithm also relies on fast matrix multiplication and runs in Mnω+o(1) time Our result shows that, at least for small integer weights, the replacement paths problem in directed graphs may be easier than the related all pairs shortest paths problem in directed graphs, as the current best runtime for the latter is Ω(n25) time even if ω = 2
TL;DR: The QVG that uses multi-level quantization, partial obstacle representation and SID visibility checking provides the shortest best path and has lower computational cost than all other methods.
Abstract: We propose variants of the quantized visibility graph (QVG) for efficient path planning. Conventional visibility graphs have been used for path planning when the obstacles are polygonal. The QVG ex...
Proceedings Article•
01 Jan 2011TL;DR: An algorithm computing a shortest path between to vertices in a square grid graph with edge weights that uses memory less than linear in the number of vertices (apart from that for storing in the input).
Abstract: We present an algorithm computing a shortest path between to vertices in a square grid graph with edge weights that uses memory less than linear in the number of vertices (apart from that for storing in the input). For any e > 0, our algorithm uses a work space of
TL;DR: It is proved that the Limited Packing problem is NP- complete for split graphs and for bipartite graphs, two graph classes for which the Multiple Domination problem is also NP-complete (Liao and Chang, 2003).
23 Jan 2011
TL;DR: In this paper, the authors investigated the complexity of shortest paths in time-dependent graphs, in which the costs of edges vary as a function of time, and as a result the shortest path between two nodes s and d can change over time.
Abstract: We investigate the complexity of shortest paths in time-dependent graphs, in which the costs of edges vary as a function of time, and as a result the shortest path between two nodes s and d can change over time. Our main result is that when the edge cost functions are (polynomial-size) piecewise linear, the shortest path from s to d can change nθ(log n) times, settling a several-year-old conjecture of Dean [Technical Reports, 1999, 2004]. We also show that the complexity is polynomial if the slopes of the linear function come from a restricted class, present an output-sensitive algorithm for the general case, and describe a scheme for a (1 + e)-approximation of the travel time function in near-quadratic space. Finally, despite the fact that the arrival time function may have superpolynomial complexity, we show that a minimum delay path for any departure time interval can be computed in polynomial time.
Journal Article•
TL;DR: Strong polynomial algorithms are obtained for a variety of path problems on acyclic and general digraphs based on the solution of resource constrained shortest path problems with equality constraints which can be solved in pseudo-polynomial time.
Abstract: The shortest path problem in which the (s, t) -paths P of a given digraph G = (V, E) are compared with respect to the sum of their edge costs is one of the best known problems in combinatorial optimization. The paper is concerned with a number of variations of this problem having different objective functions like bottleneck, balanced, minimum deviation, algebraic sum, k -sum and k -max objectives, (k 1, k 2) -max, (k 1, k 2) -balanced and several types of trimmed-mean objectives. We give a survey on existing algorithms and propose a general model for those problems not yet treated in literature. The latter is based on the solution of resource constrained shortest path problems with equality constraints which can be solved in pseudo-polynomial time if the given graph is acyclic and the number of resources is fixed. In our setting, however, these problems can be solved in strongly polynomial time. Combining this with known results on k -sum and k -max optimization for general combinatorial problems, we obtain strongly polynomial algorithms for a variety of path problems on acyclic and general digraphs.
TL;DR: The expressiveness of a positive fragment of path queries, denoted Path+, is studied to establish that Path+ is equivalent in expressive power to two particular subfragments, as well as to the class of tree queries, a subclass of the first-order conjunctive queries defined over the label, parent–child and child–parent predicates.
Abstract: We study the expressiveness of a positive fragment of path queries, denoted Path+, on documents that can be represented as node-labeled trees. The expressiveness of Path+ is studied from two angles. First, we establish that Path+ is equivalent in expressive power to two particular subfragments, as well as to the class of tree queries, a subclass of the first-order conjunctive queries defined over the label, parent–child and child–parent predicates. The translation algorithm from tree queries to Path+ yields a normal form for Path+ queries. Using this normal form, we can decompose a Path+ query into subqueries that can be expressed in a very small fragment of Path+ for which efficient evaluation strategies are available. Second, we characterize the expressiveness of Path+ in terms of its ability to resolve nodes in a document. This result is used to show that each tree query can be translated to a unique, equivalent and minimal tree query. The combination of these results yields an effective strategy to evaluate a large class of path queries on documents.
TL;DR: By incorporating the recursion into the canonical construction path method of isomorph rejection, a generator of non-isomorphic embedded 5-regular planar graphs is obtained with time complexity O(n 2 ) per isomorphism class.
Abstract: We describe how the 5-regular simple planar graphs can all be obtained from an elementary family of starting graphs by repeatedly applying a few local expansion operations. The proof uses an amalgam of theory and computation. By incorporating the recursion into the canonical construction path method of isomorph rejection, a generator of non-isomorphic embedded 5-regular planar graphs is obtained with time complexity O(n 2 ) per isomorphism class. A similar result is obtained for simple planar pentangulations with minimum degree 2.