Shortest path problem with forbidden paths: The elementary version

This paper addresses the elementary shortest path problem with forbidden paths. The main aim is to find the shortest paths from a single origin node to every other node of a directed graph, such that the solution does not contain any path belonging to a given set (i.e., the forbidden set). It is imposed that no cycle can be included in the solution. The problem at hand is formulated as a particular instance of the shortest path problem with resource constraints and two different solution approaches are defined and implemented. One is a Branch & Bound based algorithm, the other is a dynamic programming approach. Different versions of the proposed solution strategies are developed and tested on a large set of test problems.     

Shortest paths algorithms: Theory and experimental evaluation

We conduct an extensive computational study of shortest paths algorithms, including some very recent algorithms. We also suggest new algorithms motivated by the experimental results and prove interesting theoretical results suggested by the experimental data. Our computational study is based on several natural problem classes which identify strengths and weaknesses of various algorithms. These problem classes and algorithm implementations form an environment for testing the performance of shortest paths algorithms. The interaction between the experimental evaluation of algorithm behavior and the theoretical analysis of algorithm performance plays an important role in our research. This work was done while Boris V. Cherkassky was visiting Stanford University Computer Science Department and supported by the NSF and Powell Foundation grants mentioned below. Andrew V. Goldberg was supported in part by ONR Young Investigator Award N00014-91-J-1855, NSF Presidential Young Investigator Grant CCR-8858097 with matching funds from AT&T, DEC, and 3M, and a grant from Powell Foundation. Corresponding author. This work was done while Tomasz Radzik was a Postdoctoral Fellow at SORIE, Cornell University, and supported by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550, and by the Packard Fellowship of éva Tardos.     

On shortest path games

A class of cooperative TU-games arising from shortest path problems is introduced and analyzed. Some conditions under which a shortest path game is balanced are obtained. Also an axiomatic characterization of the Shapley value for this class of games is provided.     

LP formulations of the shortest path tree problem

Based on a pair of primal-dual LP formulations of the shortest path tree problem, the first algorithmic approach to reoptimizing the shortest paths subject to changes in the edge weights was proposed by S. Pallottino and M.G. Scutellá in 2003. We shall here focus solely on their introductory sections, propose some simplifications of the models considered, and finally relate the resulting models to the presentation of single-source shortest path problems in textbooks treating this subject with but limited or no reference to LP.Received: April 2004, Revised: August 2004, MSC classification: 90C05, 90C35, 90B10 Dedicated to the memory of Stefano Pallottino     

Link distance and shortest path problems in the plane

This paper describes algorithms to compute Voronoi diagrams, shortest path maps, the Hausdorff distance, and the Fréchet distance in the plane with polygonal obstacles. The underlying distance measures for these algorithms are either shortest path distances or link distances. The link distance between a pair of points is the minimum number of edges needed to connect the two points with a polygonal path that avoids a set of obstacles. The motivation for minimizing the number of edges on a path comes from robotic motions and wireless communications because turns are more difficult in these settings than straight movements.Link-based Voronoi diagrams are different from traditional Voronoi diagrams because a query point in the interior of a Voronoi face can have multiple nearest sites. Our site-based Voronoi diagram ensures that all points in a face have the same set of nearest sites. Our distance-based Voronoi diagram ensures that all points in a face have the same distance to a nearest site.The shortest path maps in this paper support queries from any source point on a fixed line segment. This is a middle-ground approach because traditional shortest path maps typically support queries from either a fixed point or from all possible points in the plane.The Hausdorff distance and Fréchet distance are fundamental similarity metrics for shape matching. This paper shows how to compute new variations of these metrics using shortest paths or link-based paths that avoid polygonal obstacles in the plane.     

Optimal cost sharing for capacitated facility location games

We consider cost sharing for a class of facility location games, where the strategy space of each player consists of the bases of a player-specific matroid defined on the set of resources. We assume that resources have nondecreasing load-dependent costs and player-specific delays. Our model includes the important special case of capacitated facility location problems, where players have to jointly pay for opened facilities. The goal is to design cost sharing protocols so as to minimize the resulting price of anarchy and price of stability. We investigate two classes of protocols: basic protocols guarantee the existence of at least one pure Nash equilibrium and separable protocols additionally require that the resulting cost shares only depend on the set of players on a resource. We find optimal basic and separable protocols that guarantee the price of stability/price of anarchy to grow logarithmically/linearly in the number of players. These results extend our previous results (cf. von Falkenhausen & Harks, 2013), where optimal basic and separable protocols were given for the case of symmetric matroid games without delays.     

Accelerated label setting algorithms for the elementary resource constrained shortest path problem

A label setting algorithm for solving the Elementary Resource Constrained Shortest Path Problem, using node resources to forbid repetition of nodes on the path, is implemented. A state-space augmenting approach for accelerating run times is considered. Several augmentation strategies are suggested and compared numerically.     

Evolutionary games

This paper deals with a mathematical game. As the name implies, the game concept is formulated with biological evolution in mind. An evolutionary game differs from the usual game concepts in that the players cannot choose their strategies. Rather, the strategies used by the players are handed down from generation to generation. It is the survival characteristics of a strategy that determine the outcome of the evolutionary game. Players interact and receive payoffs according to the strategies they are using. These interactions, in turn, determine the fitness of players using a given strategy. The survival characteristics of strategy are determined directly from the fitness functions. Necessary conditions for determining an evolutionarily stable strategy are developed here for a continuous game. Results are illustrated with an example.Dedicated to G. LeitmannThis work was supported by NSF Grant No. INT-82-10803 and The University of Western Australia (Visiting Fellowship, Department of Mathematics, 1983).     

Maximizing the minimum source-sink path subject to a budget constraint

Given a linear cost function for lengthening arcs, a technique is shown for maximizing, within a budget, the shortest source—sink path length in a graph. The computation is equivalent to the parametric solution of a minimum cost flow problem.This work was done while G.C. Harding was at Cornell University.The work of D.R. Fulkerson was supported by the National Science Foundation under Grant MPS74-24026 and by the Office of Naval Research under Grant NR 044-439.     

On the cardinality of the Pareto set in bicriteria shortest path problems

Computing shortest paths with two or more conflicting optimization criteria is a fundamental problem in transportation and logistics. We study the problem of finding all Pareto-optimal solutions for the multi-criteria single-source shortest-path problem with nonnegative edge lengths. The standard approaches are generalizations of label-setting (Dijkstra) and label-correcting algorithms, in which the distance labels are multi-dimensional and more than one distance label is maintained for each node. The crucial parameter for the run time and space consumption is the total number of Pareto optima. In general, this value can be exponentially large in the input size. However, in various practical applications one can observe that the input data has certain characteristics, which may lead to a much smaller number—small enough to make the problem efficiently tractable from a practical viewpoint. For typical characteristics which occur in various applications we study in this paper whether we can bound the size of the Pareto set to a polynomial size or not. These characteristics are also evaluated (1) on a concrete application scenario (computing the set of best train connections in view of travel time, fare, and number of train changes) and (2) on a simplified randomized model. It will turn out that the number of Pareto optima on each visited node is restricted by a small constant in our concrete application, and that the size of the Pareto set is much smaller than our worst case bounds in the randomized model. A preliminary short version of this paper appeared in the Proceedings of the 5th Workshop on Algorithm Engineering (WAE 2001), LNCS 2141, Springer Verlag, pp. 185–197 (2001) under the title “Pareto shortest paths is often feasible in practice.”     

Definitions of equilibrium in network formation games

We examine a variety of stability and equilibrium definitions that have been used to study the formation of social networks among a group of players. In particular we compare variations on three types of definitions: those based on a pairwise stability notion, those based on the Nash equilibria of a link formation game, and those based on equilibria of a link formation game where transfers are possible.Bloch is also affiliated with the University of Warwick.     

求最短路径的“改进标号法”

本给出了求赋权图中两顶点之间最短路径的“改进标号法”,该方法在效率上优于Ｄｉｊｋｓｔｒａ的标号法,并在确定最短路径的长度的同时,也确定了相应的最短路径。     

Depreciation games

The main aim of this paper is to show how Game Theory can be used in the day-to-day running of a company. No precedent, as far as we know, in the literature of the application of Cooperative Game Theory in order to obtain new depreciation methods for a fixed asset which is owned by a company. To this end, we will first illustrate the usual depreciation methods employed in accountancy. Then, with the objective of finding new depreciation methods, we will establish the relationship between Game Theory and Accountancy. On the basis of this relationship we introduce some new depreciation methods. Finally, a family of solutions is defined which improves the properties of traditional and previously introduced methods. Financial support from the Generalitat Valenciana through project GV05/077 is gratefully acknowledged by J. Aparicio. Financial support from the Generalitat Valenciana through project GRUPOS04/79 and from the Ministerio de Educación y Ciencia through project HI2002/0032 are gratefully acknowledged by J. Sánchez-Soriano.     

Economic lot-sizing games

In this paper we introduce a new class of OR games: economic lot-sizing (ELS) games. There are a number of retailers that have a known demand for a fixed number of periods. To satisfy demand the retailers order products at the same manufacturer. By placing joint orders instead of individual orders, costs can be reduced and a cooperative game arises. In this paper we show that ELS games are balanced. Furthermore, we show that two special classes of ELS games are concave.     

Complete stability of noncooperative games

This paper is concerned with a class of noncooperative games ofn players that are defined byn reward functions which depend continuously on the action variables of the players. This framework provides a realistic model of many interactive situations, including many common models in economics, sociology, engineering, and political science. The concept of Nash equilibrium is a suitable companion to such models.A variety of different sufficient conditions for existence, uniqueness, and stability of a Nash equilibrium point have been previously proposed. By sharpening the noncooperative aspect of the framework (which is really only implicit in the original framework), this paper attempts to isolate one set of natural conditions that are sufficient for existence, uniqueness, and stability. It is argued thatl quasicontraction is such a natural condition. The concept of complete stability is introduced to reflect the full character of noncooperation. It is then shown that, in the linear case, the condition ofl quasicontraction is both necessary and sufficient for complete stability.This research was supported by the Air Force Office of Scientific Research under Grant No. AFOSR 77-3141 and by the National Science Foundation under Grant No. GK18748.     

Survivable network design under optimal and heuristic interdiction scenarios

We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer’s network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.     

Virtual path layouts optimizing total hop count on ATM tree networks

This paper presents some algorithmic results concerning virtual path layouts for the one-to-many communication problem in ATM tree networks. The ATM network model is based on covering the network with a layout of virtual paths, under some constraints on the allowed load, namely, the number of paths that can share an edge. The quality measure used is the hop count, namely, the number of edges traversed between two vertices that need to communicate. Whereas most former results concerned the maximum hop count of the virtual path layout, our interest here is in measuring its total hop count, or alternatively its average hop count. The paper presents a dynamic programming algorithm for planning ATM network layouts with minimal total hop count for one-to-many requirements under load constraints over the class of tree networks.     

Solving the shortest path tour problem

In this paper, we study the shortest path tour problem in which a shortest path from a given origin node to a given destination node must be found in a directed graph with non-negative arc lengths. Such path needs to cross a sequence of node subsets that are given in a fixed order. The subsets are disjoint and may be different-sized. A polynomial-time reduction of the problem to a classical shortest path problem over a modified digraph is described and two solution methods based on the above reduction and dynamic programming, respectively, are proposed and compared with the state-of-the-art solving procedure. The proposed methods are tested on existing datasets for this problem and on a large class of new benchmark instances. The computational experience shows that both the proposed methods exhibit a consistent improved performance in terms of computational time with respect to the existing solution method.     

Equilibria of two-sided matching games with common preferences

Problems of matching have long been studied in the operations research literature (assignment problem, secretary problem, stable marriage problem). All of these consider a centralized mechanism whereby a single decision maker chooses a complete matching which optimizes some criterion. This paper analyzes a more realistic scenario in which members of the two groups (buyers–sellers, employers–workers, males–females) randomly meet each other in pairs (interviews, dates) over time and form couples if there is mutual agreement to do so. We assume members of each group have common preferences over members of the other group. Generalizing an earlier model of Alpern and Reyniers [Alpern, S., Reyniers, D.J., 2005. Strategic mating with common preferences. J. Theor. Biol. 237, 337–354], we assume that one group (called males) is r   times larger than the other, r?1$r?1$. Thus all females, but only 1/r$1/r$ of the males, end up matched. Unmatched males have negative utility -c$-c$. We analyze equilibria of this matching game, depending on the parameters r   and c$c$. In a region of (r,c)$(r\text{,}c)$ space with multiple equilibria, we compare these, and analyze their ‘efficiency’ in several respects. This analysis should prove useful for designers of matching mechanisms who have some control over the sex ratio (e.g. by capping numbers of males at a ‘singles event’or by having ‘ladies free’ nights) or the nonmating cost c (e.g. tax benefits to married couples).     

Time version of the shortest path problem in a stochastic-flow network

Many studies on hardware framework and routing policy are devoted to reducing the transmission time for a flow network. A time version of the shortest path problem thus arises to find a quickest path, which sends a given amount of data from the unique source to the unique sink with minimum transmission time. More specifically, the capacity of each arc in the flow network is assumed to be deterministic. However, in many real-life networks, such as computer systems, telecommunication systems, etc., the capacity of each arc should be stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Hence, the minimum transmission time is not a fixed number. We extend the quickest path problem to evaluating the probability that d$d$ units of data can be sent under the time constraint T$T$. Such a probability is named the system reliability. In particular, the data are transmitted through two minimal paths simultaneously in order to reduce the transmission time. A simple algorithm is proposed to generate all (d,T)$(d,T)$-MPs and the system reliability can then be computed in terms of (d,T)$(d,T)$-MPs. Moreover, the optimal pair of minimal paths with highest system reliability could be obtained.