Enumerating K best paths in length order in DAGs

We address the problem of finding the K best paths connecting a given pair of nodes in a directed acyclic graph (DAG) with arbitrary lengths. One of the main results in this paper is the proof that a tree representing the kth shortest path is obtained by an arc exchange in one of the previous (k − 1) trees (each of which contains a previous best path). An O(m + K(n + log K)) time and O(K + m) space algorithm is designed to explicitly determine the K shortest paths in a DAG with n nodes and m arcs. The algorithm runs in O(m + Kn) time using O(K + m) space in DAGs with integer length arcs. Empirical results confirming the superior performance of the algorithm to others found in the literature for randomly generated graphs are reported.     

Isometric path numbers of graphs

An isometric path between two vertices in a graph G is a shortest path joining them. The isometric path number of G, denoted by ip(G), is the minimum number of isometric paths needed to cover all vertices of G. In this paper, we determine exact values of isometric path numbers of complete r-partite graphs and Cartesian products of 2 or 3 complete graphs.     

The shortest path problem with forbidden paths

We consider a variant of the constrained shortest path problem, where the constraints come from a set of forbidden paths (arc sequences) that cannot be part of any feasible solution. Two solution approaches are proposed for this variant. The first uses Aho and Corasick's keyword matching algorithm to filter paths produced by a k-shortest paths algorithm. The second generalizes Martins' deviation path approach for the k-shortest paths problem by merging the original graph with a state graph derived from Aho and Corasick's algorithm. Like Martins' approach, the second method amounts to a polynomial reduction of the shortest path problem with forbidden paths to a classic shortest path problem. Its significant advantage over the first approach is that it allows considering forbidden paths in more general shortest path problems such as the shortest path problem with resource constraints.     

On the Hardness of Equal Shortest Path Routing

In telecommunication networks packets are carried from a source s to a destination t on a path that is determined by the underlying routing protocol. Most routing protocols belong to the class of shortest path routing protocols. In such protocols, the network operator assigns a length to each link. A packet going from s to t follows a shortest path according to these lengths. For better protection and efficiency, one wishes to use multiple (shortest) paths between two nodes. Therefore the routing protocol must determine how the traffic from s to t is distributed among the shortest paths. In the protocol called OSPF-ECMP (for Open Shortest Path First-Equal Cost Multiple Path) the traffic incoming at every node is uniformly balanced on all outgoing links that are on shortest paths. In that context, the operator task is to determine the “best” link lengths, toward a goal such as maximizing the network throughput for given link capacities.In this work, we show that the problem of maximizing even a single commodity flow for the OSPF-ECMP protocol cannot be approximated within any constant factor ratio. Besides this main theorem, we derive some positive results which include polynomial-time approximations and an exponential-time exact algorithm. We also prove that despite their weakness, our approximation and exact algorithms are, in a sense, the best possible.     

Best piecewise constant approximation of a function of single variable

This paper is concerned with the best piecewise constant approximation of a function f of single variable. Polynomial time algorithms are derived by using shortest path and dynamic programming techniques. Several applications of this class of problems will be briefly touched upon.     

The directional p-median problem: Definition,complexity, and algorithms

An instance of a p-median problem gives n demand points. The objective is to locate p supply points in order to minimize the total distance of the demand points to their nearest supply point. p-Median is polynomially solvable in one dimension but NP-hard in two or more dimensions, when either the Euclidean or the rectilinear distance measure is used. In this paper, we treat the p-median problem under a new distance measure, the directional rectilinear distance, which requires the assigned supply point for a given demand point to lie above and to the right of it. In a previous work, we showed that the directional p-median problem is polynomially solvable in one dimension; we give here an improved solution through reformulating the problem as a special case of the constrained shortest path problem. We have previously proven that the problem is NP-complete in two or more dimensions; we present here an efficient heuristic to solve it. Compared to the robust Teitz and Bart heuristic, our heuristic enjoys substantial speedup while sacrificing little in terms of solution quality, making it an ideal choice for real-world applications with thousands of demand points.     

On the convergence of finite state mean-field games through Γ-convergence

In this study, we consider the long-term convergence (trend toward an equilibrium) of finite state mean-field games using Γ-convergence. Our techniques are based on the observation that an important class of mean-field games can be viewed as the Euler–Lagrange equation of a suitable functional. Therefore, using a scaling argument, one can convert a long-term convergence problem into a Γ-convergence problem. Our results generalize previous results related to long-term convergence for finite state problems.     

Verifying a p-adic abelian Stark conjecture at s=1

In a previous paper (Ann. L’ Inst. Fourier 52(2) (2002) 379-417) the second-named author developed a new approach to the abelian p-adic Stark conjecture at s=1 and stated some related conjectures. The aim of the present paper is to develop and apply techniques to numerically investigate one of these—the ‘Weak Refined Combined Conjecture’—in 15 cases.     

A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games

We suggest the first strongly subexponential and purely combinatorial algorithm for solving the mean payoff games problem. It is based on iteratively improving the longest shortest distances to a sink in a possibly cyclic directed graph.We identify a new “controlled” version of the shortest paths problem. By selecting exactly one outgoing edge in each of the controlled vertices we want to make the shortest distances from all vertices to the unique sink as long as possible. The decision version of the problem (whether the shortest distance from a given vertex can be made bigger than a given bound?) belongs to the complexity class NP∩CONP. Mean payoff games are easily reducible to this problem. We suggest an algorithm for computing longest shortest paths. Player MAX selects a strategy (one edge from each controlled vertex) and player MIN responds by evaluating shortest paths to the sink in the remaining graph. Then MAX locally changes choices in controlled vertices looking at attractive switches that seem to increase shortest paths lengths (under the current evaluation). We show that this is a monotonic strategy improvement, and every locally optimal strategy is globally optimal. This allows us to construct a randomized algorithm of complexity , which is simultaneously pseudopolynomial (W is the maximal absolute edge weight) and subexponential in the number of vertices n. All previous algorithms for mean payoff games were either exponential or pseudopolynomial (which is purely exponential for exponentially large edge weights).     

A class of graphs for which the ulam conjecture holds

A median graph is a connected graph, such that for any three vertices u,ν and w there is exactly one vertex x that lies simultaneously on a shortest (u, ν)-path a shortest (ν, w)-path and a shortest (w, u)-path. It is proved that a median graph can be obtained from a one-vertex graph by an expansion procedure. From this characterization some nice properties are derived.     

A trust region method for solving the decentralized static output feedback design problem

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents anO(n ²) time sequential algorithm and anO(n ²/p+logn) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, wherep andn represent respectively the number of processors and the number of vertices of the circular-arc graph.     

On the second point-to-point undirected shortest simple path problem

We address the determination of the second point-to-point shortest simple path in undirected networks. The effective reduced cost concept is introduced to compute the second best solution. This concept is used to prove that a path tree containing the second point-to-point shortest simple path is adjacent to any shortest path tree. Therefore, this result immediately implies a method requiring O(m) time once that the shortest path tree is obtained on an undirected network with n nodes and m edges.     

On properties of discrete (r, q) and (s, T) inventory systems

We consider single-item (r, q) and (s, T) inventory systems with integer-valued demand processes. While most of the inventory literature studies continuous approximations of these models and establishes joint convexity properties of the policy parameters in the continuous space, we show that these properties no longer hold in the discrete space, in the sense of linear interpolation extension and L^?-convexity. This nonconvexity can lead to failure of optimization techniques based on local optimality to obtain the optimal inventory policies. It can also make certain comparative properties established previously using continuous variables invalid. We revise these properties in the discrete space.     

Mathematical programming approaches for generating p-efficient points

Probabilistically constrained problems, in which the random variables are finitely distributed, are non-convex in general and hard to solve. The p-efficiency concept has been widely used to develop efficient methods to solve such problems. Those methods require the generation of p-efficient points (pLEPs) and use an enumeration scheme to identify pLEPs. In this paper, we consider a random vector characterized by a finite set of scenarios and generate pLEPs by solving a mixed-integer programming (MIP) problem. We solve this computationally challenging MIP problem with a new mathematical programming framework. It involves solving a series of increasingly tighter outer approximations and employs, as algorithmic techniques, a bundle preprocessing method, strengthening valid inequalities, and a fixing strategy. The method is exact (resp., heuristic) and ensures the generation of pLEPs (resp., quasi pLEPs) if the fixing strategy is not (resp., is) employed, and it can be used to generate multiple pLEPs. To the best of our knowledge, generating a set of pLEPs using an optimization-based approach and developing effective methods for the application of the p-efficiency concept to the random variables described by a finite set of scenarios are novel. We present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques.     

Some heuristic methods for solving p-median problems with a coverage constraint

The aim of this paper is to solve p-median problems with an additional coverage constraint. These problems arise in location applications, when the trade-off between distance and coverage is being calculated. Three kinds of heuristic algorithms are developed. First, local search procedures are designed both for constructing and improving feasible solutions. Second, a multistart GRASP heuristic is developed, based on the previous local search methods. Third, by employing Lagrangean relaxation methods, a very efficient Lagrangean heuristic algorithm is designed, which extends the well known algorithm of Handler and Zang, for constrained shortest path problems, to constrained p-median problems. Finally, a comparison of the computational efficiency of the developed methods is made between a variety of problems of different sizes.     

An extension of labeling techniques for finding shortest path trees

Label setting techniques are all based on Dijkstra’s condition of always scanning the node with the minimum label, which guarantees that each node will be scanned exactly once; while this condition is sufficient it is not necessary. In this paper, we discuss less restrictive conditions that allow the scanning of a node that does not have the minimum label, yet still maintaining sufficiency in scanning each node exactly once; various potential shortest path schemes are discussed, based on these conditions. Two approaches, a label setting and a flexible hybrid one are designed and implemented. The performance of the algorithms is assessed both theoretically and computationally. For comparative analysis purposes, three additional shortest path algorithms – the commonly cited in the literature – are coded and tested. The results indicate that the approaches that rely on the less restrictive optimality conditions perform substantially better for a wide range of network topologies.     

Self-improving properties of inequalities of Poincaré type on measure spaces and applications

We show that the self-improving nature of Poincaré estimates persists for domains in rather general measure spaces. We consider both weak type and strong type inequalities, extending techniques of B. Franchi, C. Pérez and R. Wheeden. As an application in spaces of homogeneous type, we derive global Poincaré estimates for a class of domains with rough boundaries that we call ?-John domains, and we show that such domains have the requisite properties. This class includes John (or Boman) domains as well as s-John domains. Further applications appear in a companion paper.     

Multigrid solvers in reconfigurable hardware

The problem of finding the solution of partial differential equations (PDEs) plays a central role in modeling real world problems. Over the past years, Multigrid solvers have showed their robustness over other techniques, due to its high convergence rate which is independent of the problem size. For this reason, many attempts for exploiting the inherent parallelism of Multigrid have been made to achieve the desired efficiency and scalability of the method. Yet, most efforts fail in this respect due to many factors (time, resources) governed by software implementations. In this paper, we present a hardware implementation of the V-cycle Multigrid method for finding the solution of a 2D-Poisson equation. We use Handel-C to implement our hardware design, which we map onto available field programmable gate arrays (FPGAs). We analyze the implementation performance using the FPGA vendor's tools. We demonstrate the robustness of Multigrid over other similar iterative solvers, such as Jacobi and successive over relaxation (SOR  ), in both hardware and software. We compare our findings with a C++$\mathrm{C}++$ version of each algorithm. The obtained results show better performance when compared to existing software versions.