Finding all nondominated points of multi-objective integer programs

We develop exact algorithms for multi-objective integer programming (MIP) problems. The algorithms iteratively generate nondominated points and exclude the regions that are dominated by the previously-generated nondominated points. One algorithm generates new points by solving models with additional binary variables and constraints. The other algorithm employs a search procedure and solves a number of models to find the next point avoiding any additional binary variables. Both algorithms guarantee to find all nondominated points for any MIP problem. We test the performance of the algorithms on randomly-generated instances of the multi-objective knapsack, multi-objective shortest path and multi-objective spanning tree problems. The computational results show that the algorithms work well.     

结点有约束的交通网络最短路径模型

结点有约束的网络是一类特殊的网络，如具有禁止通行限制信息的交通路网等，由于最短路径的求解是有后效性的，经典的Dijkstra算法等不能直接用来求解该问题，本文提出了一种结点有约束的交通网络最短路径建模方法，该方法所建模型为一般网络模型，可用任一传统高效的算法求其最短路径，从根本上降低了问题的复杂性，为很好地解决交通、通信等领域中的此类问题提供了有益的方法。     

An evaluation of mathematical programming and minicomputers

The availability of efficient mathematical software on minicomputers could greatly increase the use of operations research techniques in industry and government. The objective of this paper is to demonstrate the feasibility of implementing a particular class of mathematical programming algorithms, namely shortest path algorithms, on “typical” minicomputers. Two distinct shortest path algorithms were tested on four computer systems using a common set of test problems. Computational results are presented which verify the feasibility of implementing these algorithms in a minicomputer environment, and also show the relative efficiency of each algorithm to be the same when tested on a minicomputer as when tested on a large-scale computer system.     

An extended shortest path problem: A data envelopment analysis approach

A special and important network structured linear programming problem is the shortest path problem. Classical shortest path problems assume that there are unit of shipping cost or profit along an arc. In many real occasions, various attributes (various costs and profits) are usually considered in a shortest path problem. Because of the frequent occurrence of such network structured problems, there is a need to develop an efficient procedure for handling these problems. This paper studies the shortest path problem in the case that multiple attributes are considered along the arcs. The concept of relative efficiency is defined for each path from initial node to final node. Then, an efficient path with the maximum efficiency is determined.     

Improved algorithms for replacement paths problems in restricted graphs

We present near-optimal algorithms for two problems related to finding the replacement paths for edges with respect to shortest paths in sparse graphs. The problems essentially study how the shortest paths change as edges on the path fail, one at a time. Our technique improves the existing bounds for these problems on directed acyclic graphs, planar graphs, and non-planar integer-edge-weighted graphs.     

Computing shortest paths in networks derived from recurrence relations

Dynamic programming formulations of optimization problems often call for the computation of shortest paths in networks derived from recurrence relations. These derived networks tend to be very large, but they are also very regular and lend themselves to the computation of nontrivial lower bounds on path lengths. In this tutorial paper, we describe unidirectional and bidirectional search procedures that make use of bounding information in computing shortest paths. When applied to many optimization problems, these shortest path algorithms capture the advantages of both dynamic programming and branch-and-bound.     

机器人避障问题

研究机器人在平面区域中绕过静态障碍物到达指定目的地的问题, 分别考虑了路程最短和时间最短两种目标下的最优路径, 给出了计算机自动搜索最优路径的模型和算法。     

Path problems in skew-symmetric graphs

We study path problems in skew-symmetric graphs. These problems generalize the standard graph reachability and shortest path problems. We establish combinatorial solvability criteria and duality relations for the skew-symmetric path problems and use them to design efficient algorithms for these problems. The algorithms presented are competitive with the fastest algorithms for the standard problems.This research was done while the first author was at Stanford University Computer Science Department, supported in part by ONR Office of Naval Research 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.This research was done while the second author was visiting Stanford University Computer Science Department and supported by the above mentioned NSF and Powell Foundation Grants.     

带区间数据的最小风险斯坦纳树问题

考虑了在带区间数据的不确定网络中, 最小风险和模型以及最小最大风险模型下的斯坦纳树问题. 它们推广了相应模型下的最短路问题和最小支撑树问题, 在网络设计中具有更加广泛的应用.我们分别给出了这两个模型下斯坦纳树问题的近似算法, 并对算法性能做了理论分析和证明. 结果显示我们的算法具有优良的常数逼近的性质, 能在多项式时间内算出令人满意的解.     

Exact algorithms for the stochastic shortest path problem with a decreasing deadline utility function

In this paper, the stochastic shortest path problem of determining a path that maximizes the expected utility is considered. The nature of the utility function used to evaluate paths is of a decreasing deadline type. The principal contribution of this paper is the development of exact algorithms that use two types of pruning techniques that are incorporated in labeling procedures. One type of pruning makes use of the concept of local preference relations while the other type makes use of relaxations. Specifically two algorithms are developed, each containing the same preference relation, but two different relaxations. Our extensive computational testing indicate that both these algorithms are able to solve even large size problems quickly. More importantly, even for large problems, both the algorithms solved them by enumerating a very small percentage of all paths.     

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.     

A heuristic approach to hard constrained shortest path problems

Constrained shortest path problems have many applications in areas like network routing, investments planning and project evaluation as well as in some classical combinatorial problems with high duality gaps where even obtaining feasible solutions is a difficult task in general.We present in this paper a systematic method for obtaining good feasible solutions to hard (doubly constrained) shortest path problems. The algorithm is based essentially on the concept of efficient solutions which can be obtained via parametric shortest path calculations. The computational results obtained show that the approach proposed here leads to optimal or very good near optimal solutions for all the problems studied.From a theoretical point of view, the most important contribution of the paper is the statement of a pseudopolynomial algorithm for generating the efficient solutions and, more generally, for solving the parametric shortest path problem.     

The robust shortest path problem with interval data via Benders decomposition

Many real problems can be modelled as robust shortest path problems on digraphs with interval costs, where intervals represent uncertainty about real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs. In this paper we discuss the application of a Benders decomposition approach to this problem. Computational results confirm the efficiency of the new algorithm. It is able to clearly outperform state-of-the-art algorithms on many classes of networks. For the remaining classes we identify the most promising algorithm among the others, depending of the characteristics of the networks. Received: September 2004 / Accepted: March 2005 AMS classification: 90C47, 52B05, 90C57 The work was partially supported by the European Commission IST projects MOSCA (IST-2000-29557) and BISON (IST-2001-38923). All correspondence to: Roberto Montemanni     

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.     

A preference-based approach to spanning trees and shortest paths problems****

Comparison of solutions in combinatorial problems is often based on an additive cost function inducing a complete order on solutions. We investigate here a generalization of the problem, where preferences take the form of a quasi-transitive binary relation defined on the solutions space. We first propose preference-based search algorithms for two classical combinatorial problems, namely the preferred spanning trees problem (a generalization of the minimum spanning tree problem) and the preferred paths problem (a generalization of the shortest path problem). Then, we introduce a very useful axiom for preference relations called independence. Using this axiom, we establish admissibility results concerning our preference-based search algorithms. Finally, we address the problem of dealing with non-independent preference relations and provide different possible solutions for different particular problems (e.g. lower approximation of the set of preferred solutions for multicriteria spanning trees problems, or relaxation of the independence axiom for interval-valued preferred path problems).     

所有点对之间最快路问题

所有点对之间最快路问题就是要在所有点对(Vs，Vt)之间传送数据δs,t,并找出一条最快的路线．解决所有点对之间最快路问题的关键是产生有效解的等价集合．运用动态点对最短路的算法，本文首先设计了一个时间复杂性为O(mn^2)的产生有效解等价集合的算法，然后研究了静态点对之间最快路问题和动态点对之间最快路问题，其算法的时间复杂性分别为O(mn^2)和O(m^2n^2)．最后本文研究了求和对最小的路问题，证明该问题可以在O(mn^2)时间内解决．     

Criss-cross methods: A fresh view on pivot algorithms

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upon.     

Metric location problems with nonshortest service routes

We study the generalizations of the classical metric location problems in which the route of a service team passes through a remote depository containing some components needed for replacement. Generally speaking, in this case the path to the client is not shortest and the problem ceases to be metric. We show that the available approximation algorithms for the classical metric problems translate to our generalizations with preservation of approximation ratios.     

Computational experiments with a lazy version of a <Emphasis Type="Italic">K</Emphasis> quickest simple path ranking algorithm

The quickest path problem is related to the classical shortest path problem, but its objective function concerns the transmission time of a given amount of data throughout a path, which involves both cost and capacity. The K-quickest simple paths problem generalises the latter, by looking for a given number K of simple paths in non-decreasing order of transmission time. Two categories of algorithms are known for ranking simple paths according to the transmission time. One is the adaptation of deviation algorithms for ranking shortest simple paths (Pascoal et al. in Comput. Oper. Res. 32(3):509–520, 2005; Rosen et al. in Comput. Oper. Res. 18(6):571–584, 1991), and another is based on ranking shortest simple paths in a sequence of networks with fixed capacity lower bounds (Chen in Inf. Process. Lett. 50:89–92, 1994), and afterwards selecting the K quickest ones. After reviewing the quickest path and the K-quickest simple paths problems we describe a recent algorithm for ranking quickest simple paths (Pascoal et al. in Ann. Oper. Res. 147(1):5–21, 2006). This is a lazy version of Chen’s algorithm, able to interchange the calculation of new simple paths and the output of each k-quickest simple path. Finally, the described algorithm is computationally compared to its former version, as well as to deviation algorithms.      

On a multicriteria shortest path problem

Multicriteria shortest path problems have not been treated intensively in the specialized literature, despite their potential applications. In fact, a single objective function may not be sufficient to characterize a practical problem completely. For instance, in a road network several parameters (as time, cost, distance, etc.) can be assigned to each arc. Clearly, the shortest path may be too expensive to be used. Nevertheless the decision-maker must be able to choose some solution, possibly not the best for all the criteria.In this paper we present two algorithms for this problem. One of them is an immediate generalization of the multiple labelling scheme algorithm of Hansen for the bicriteria case. Based on this algorithm, it is proved that any pair of nondominated paths can be connected by nondominated paths. This result is the support of an algorithm that can be viewed as a variant of the simplex method used in continuous linear multiobjective programming. A small example is presented for both algorithms.