Auction algorithms for network flow problems: A tutorial introduction

This paper surveys a new and comprehensive class of algorithms for solving the classical linear network flow problem and its various special cases such as shortest path, max-flow, assignment, transportation, and transhipment problems. The prototype method, from which the other algorithms can be derived, is the auction algorithm for the assignment problem. This is an intuitive method that operates like a rel auction where persons compete for objects by raising their prices through competitive bidding; the prices can be viewed as dual variables. Conceptually, auction algorithms represent a significant departure from the cost improvement idea that underlies primal simplex and dual ascent methods; at any one iteration, they may deteriorate both the primal and the dual cost. Auction algorithms perform very well for several important types of problems, both in theory and in practice, and they are also well suited for parallel computation.     

Towards auction algorithms for large dense assignment problems

In this paper, we focus on the problem of solving large-scale instances of the linear sum assignment problem by auction algorithms. We introduce a modified auction algorithm, called look-back auction algorithm, which extends the forward auction algorithm by the ability of reusing information from previous bids. We show that it is able to reuse information from the previous bids with high efficiency for all tested types of input instances. We discuss then the design and implementation of a suite of sequential and distributed memory auction algorithms on a Linux cluster with the evaluation on several types of input instances of the linear sum assignment problem. Our results show that the look-back auction algorithm solves sequentially nearly all types of dense instances faster than other evaluated algorithms and it is more stable than the forward-reverse auction algorithm for sparse instances. Our distributed memory auction algorithms are fully memory scalable. This research has been supported by IGA CTU under grant CTU0308013 and under research program MSMT 6840770014.     

A comparison of two algorithms for the assignment problem

State-of-the-art computational results have shown that the shortest augmenting path (SAP) methods are more efficient than other primal-dual and primal-simplex based methods for solving the linear assignment problem on uniprocessor computers. There is, however, some controversy concerning their merits when compared with Bertsekas' auction algorithm on multiprocessor computers. In this study we investigate the performance of these competing methods on the Alliant FX/8. For each method, theoretical motivation, sources of parallelism and computational results are presented.     

An in-core/out-of-core method for solving large scale assignment problems

We describe how the shortest augmenting path method can be used as basis for a so called in-core/out-of-core approach for solving large assignment problems in which the data cannot be kept in central memory of a computer. Here we start by solving the assignment problem on a sparse subgraph and then we introduce the remaining edges in an outpricing/reoptimization phase. We introduce several strategies which enable to keep the working subgraph sparse throughout the procedure and even lead to an all in-core code which is faster than the basic shortest augmenting path code.

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Zusammenfassung Wir beschreiben einen sogenannten In-Core/Out-of-Core Ansatz auf der Basis der kürzesten erweiternden Wege Methode für die Lösung gro\er Zuordnungsprobleme, für die die gesamte Kostenmatrix nicht im Zentralspeicher des Rechners gehalten werden kann. Bei diesem Ansatz wird in einer ersten Phase ein Zuordnungsproblem über einem dünnen Teilgraph optimal gelöst. In einer zweiten Phase werden dann die nicht berücksichtigen Kanten mittels der optimalen Duallösung bewertet (outpricing) und gegebenenfalls eine Reoptimierung durchgeführt. Durch Anwendung spezieller Strategien wird es möglich, während der gesamten Lösung den im Zentralspeicher abzuspeichernden Teilgraphen dünn zu halten. Weiterhin zeigt sich, da\ dieser Ansatz zu einem neuen Verfahren führt, das der zugrunde liegenden kürzesten erweiternden Wege Methode überlegen ist.

     

A data parallel augmenting path algorithm for the dense linear many-to-one assignment problem

The purpose of this study is to describe a data parallel primal-dual augmenting path algorithm for the dense linear many-to-one assignment problem also known as semi-assignment. This problem could for instance be described as assigning n persons to m(n) job groups.The algorithm is tailored specifically for massive SIMD parallelism and employs, in this context, a new efficient breadth-first-search augmenting path technique which is found to be faster than the shortest augmenting path search normally used in sequential algorithms for this problem. We show that the best known sequential computational complexity of O(mn ² ) for dense problems, is reduced to the parallel complexity of O(mn), on a machine with n processors supporting reductions in O(1) time. The algorithm is easy to implement efficiently on commercially available massively parallel computers. A range of numerical experiments are performed on a Connection Machine CM200 and a MasPar MP-2. The tests show the good performance of the proposed algorithm.     

An efficient cost scaling algorithm for the assignment problem

The cost scaling push-relabel method has been shown to be efficient for solving minimum-cost flow problems. In this paper we apply the method to the assignment problem and investigate implementations of the method that take advantage of assignment's special structure. The results show that the method is very promising for practical use.This author's research 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 the Powell Foundation.This author's research was supported by the above-mentioned ONR and NSF grants.     

Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path.     

Algorithms for time-dependent bicriteria shortest path problems

In this paper we generalize the classical shortest path problem in two ways. We consider two objective functions and time-dependent data. The resulting problem, called the time-dependent bicriteria shortest path problem (TdBiSP), has several interesting practical applications, but has not gained much attention in the literature. After reviewing relevant literature we develop a new algorithm for the TdBiSP with non-negative data. Numerical tests show the superiority of our algorithm compared with an existing algorithm in the literature. Furthermore, we discuss algorithms for the TdBiSP with negative travel times and costs.     

Constructive bounds and exact expectations for the random assignment problem

The random assignment problem is to choose a minimum‐cost perfect matching in a complete n×n bipartite graph, whose edge weights are chosen randomly from some distribution such as the exponential distribution with mean 1. In this case it is known that the expectation does not grow unboundedly with n, but approaches some limiting value c* between 1.51 and 2. The limit is conjectured to be π²/6, while a recent conjecture is that for finite n, the expected cost is ∑ 1/i². This paper contains two principal results. First, by defining and analyzing a constructive algorithm, we show that the limiting expectation is c*<1.94. Second, we extend the finite‐n conjecture to partial assignments on complete m×n bipartite graphs and prove it in some limited cases. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 113–144, 1999     

Investigation of polynomial algorithms for solving the three-index planar assignment problem

Under certain additional conditions imposed on the coefficients of the objective function in the three-index planar assignment problem, a large series of computational experiments aimed at the investigation of four polynomial algorithms for finding an asymptotically optimal solution of this problem is conducted.     

Edge-swapping algorithms for the minimum fundamental cycle basis problem

We consider the problem of finding a fundamental cycle basis with minimum total cost in an undirected graph. This problem is NP-hard and has several interesting applications. Since fundamental cycle bases correspond to spanning trees, we propose a local search algorithm, a tabu search and variable neighborhood search in which edge swaps are iteratively applied to a current spanning tree. We also present a mixed integer programming formulation of the problem whose linear relaxation yields tighter lower bounds than other known formulations. Computational results obtained with our algorithms are compared with those from the best available constructive heuristic on several types of graphs. This article extends the conference paper (Amaldi et al. 2004).     

A comparison of heuristic best-first algorithms for bicriterion shortest path problems

A variety of algorithms have been proposed to solve the bicriterion shortest path problem. This article analyzes and compares the performance of three best-first (label-setting) algorithms that accept heuristic information to improve efficiency. These are NAMOA^∗, MOA^∗, and Tung & Chew’s algorithm (TC). A set of experiments explores the impact of heuristic information in search efficiency, and the relative performance of the algorithms. The analysis reveals that NAMOA^∗ is the best option for difficult problems. Its time performance can benefit considerably from heuristic information, though not in all cases. The performance of TC is similar but somewhat worse. However, the time performance of MOA^∗ is found to degrade considerably with the use of heuristic information in most cases. Explanations are provided for these phenomena.     

平衡和不平衡运输问题与分配问题的通用迭代算法

将不平衡运输问题转化成网络最短路问题,利用Floyd算法规则,给出了一种既可以解平衡和不平衡运输问题,又可以解平衡和不平衡分配问题的通用迭代算法。与专门用于解运输问题的闭合回路法和专门用于解分配问题的匈牙利法相比,这种算法不但具有通用的优点,而且更便于在计算机上运行。     

分派问题一种标号算法

文章采用一定技巧,把求最短路的Ｄｉｊｋｓｔｒａ算法用于求解分派问题,得到一种标号算法,计算复杂性仅为Ｏ（ｎ２）,比以往的算法减少了一个数量阶Ｏ（ｎ）。     

求解运输问题的一种新算法

本文将求解分派问题的标号算法成功地用于运输问题,并证明其中的非负处理可以省略,从而把Dijk-stra算法扩展到可能出现负边权的运输问题。与通常方法比较,这种方法具有直观、简单、计算量少、及易于推广等优点;最后证明该算法是多项式的,计算复杂性仅为o(n3)(当m≤n时)。     

A scaling limit for the length of the longest cycle in a sparse random digraph

We discuss the length of the longest directed cycle in the sparse random digraph , constant. We show that for large there exists a function such that a.s. The function where is a polynomial in . We are only able to explicitly give the values , although we could in principle compute any .     

Performance analysis of linear and nonlinear techniques for automatic scaling of discretized control problems

This paper introduces three (one linear and two nonlinear) automatic scaling techniques for NLPs with states and constraints spread over several orders of magnitude, without requiring complex off-the-shelf external tools. All of these methods have been compared to standard techniques and applied to three problems using SNOPT and IPOPT. The results confirm that the proposed techniques significantly improve the NLP conditioning, yielding more reliable and in some cases, faster NLP solutions.     

Genetic algorithms for portfolio selection problems with minimum transaction lots

Conventionally, portfolio selection problems are solved with quadratic or linear programming models. However, the solutions obtained by these methods are in real numbers and difficult to implement because each asset usually has its minimum transaction lot. Methods considering minimum transaction lots were developed based on some linear portfolio optimization models. However, no study has ever investigated the minimum transaction lot problem in portfolio optimization based on Markowitz’ model, which is probably the most well-known and widely used. Based on Markowitz’ model, this study presents three possible models for portfolio selection problems with minimum transaction lots, and devises corresponding genetic algorithms to obtain the solutions. The results of the empirical study show that the portfolios obtained using the proposed algorithms are very close to the efficient frontier, indicating that the proposed method can obtain near optimal and also practically feasible solutions to the portfolio selection problem in an acceptable short time. One model that is based on a fuzzy multi-objective decision-making approach is highly recommended because of its adaptability and simplicity.