Ulrich Derigs

Implementing Goldberg's max-flow-algorithm — A computational investigation

In this paper we review Goldbergs algorithm for solving max-flow-problems on networks and we discuss several ideas for implementing and enhancing this approach. We present computational results for these implementations and we compare our best Goldberg-codes with implementations of Dinics and Karzanovs method presented in literature.Our results show that the implementations of Goldbergs algorithm outperform the other codes significantly. Here the essential breakthrough is obtained by the use of a modification of Goldbergs basic labeling scheme by which the number of labeling steps is reduced drastically.ZusammenfassungWir diskutieren einige Strategien und Datenstrukturen für eine effiziente Implementierung des Preflow-Push-Algorithmus von Goldberg zur Bestimmung maximaler Flüsse in Netzwerken. Wir berichten über numerische Erfahrungen mit diesen Implementierungen und Vergleichtests mit publizierten Codes zur Lösung des Max-Flow-Problems, die auf der Basis der Algorithmen von Dinic und Karzanov entwickelt wurden.Unsere Ergebnisse zeigen, dafß die Goldberg-Codes wesentlich weniger Rechenzeit benötigen. Der entscheidende Durchbruch wird dabei durch eine neue Modifikation der Markierungsmethode von Goldberg erzielt, durch die die Anzahl der Markierungsschritte drastisch reduziert wird.

The shortest augmenting path method for solving assignment problems — Motivation and computational experience

In this paper we discuss the shortest augmenting path method for solving assignment problems in the following respect:we introduce this basic concept using matching theorywe present several efficient labeling techniques for constructing shortest augmenting pathswe show the relationship of this approach to several classical assignment algorithmswe present extensive computational experience for complete problems, andwe show how postoptimal analysis can be performed using this approach and naturally leads to a new, highly efficient hybrid approach for solving large-scale dense assignment problems

On the use of optimal fractional matchings for solving the (integer) matching problem

We show how optimal fractional matchings can be used to start the shortest augmenting path method for solving the (integer) matching problem. Computational results are presented which indicate that this start procedure is highly efficient, i.e. it is fast and reduces the amount of work for the shortest augementing path method significantly such that the overall computing time is reduced drastically.ZusammenfassungWir zeigen wie optimale „Fractional Matchings”, d. h. optimale Lösungen der LP-Relaxation des Matching-Problems, benutzt werden können, um Ausgangslösungen für die kürzeste erweiternde Wege-Methode zur Lösung des Matching-Problems zu konstruieren. Numerische Untersuchungen zeigen, daß diese Startprozedur höchst effizient ist und den Aufwand für den eigentlichen Matching-Algorithmus signifikant reduziert, so daß die Gesamtrechenzeit drastisch reduziert wird.

Matching problems with generalized upper bound side constraints

In this article, we develop and compare procedures for the approximate solution of weighted nonbipartite matching problems with generalized upper bound side constraints. The approaches we consider are all based on Lagrangean relaxation and dual ascent. We also use a knapsack-based procedure for finding improved feasible solutions and a “k-best” solution enumeration procedure to guarantee optimality. Our computational experiments addressed two issues: the choice of the best combination of a matching code and postoptimality routine and the choice of a dual ascent rule. Our recommended combination of procedures consistently produced solutions with a very small deviation from optimality without having to resort to the enumeration procedure.

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.ZusammenfassungWir 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.

An efficient labeling technique for solving sparse assignment problems

We describe a new implementation of the shortest augmenting path approach for solving sparse assignment problems and report computational experience documenting its efficiency.ZusammenfassungWir beschreiben eine neue Implementierung der kürzesten-erweiternden-Wege-Methode zur Lösung dünner Zuordnungsprobleme und berichten über numerische Untersuchungen, die die Effizienz dieser Implementierung dokumentieren.

The Hitchcock Transportation Problem

In this section we discuss the second key problem within the class of general matching problems — the Hitchcock transportation problem (HTP). We have already shown that this problem can be viewed either as the specialization of the min-cost-flow problem on a directed bipartite graph or the specialization of the b-matching problem on a (nondirected) bipartite graph.

Near Equivalence of Network Flow Algorithms

So far we have introduced the most common algorithms for solving network flow problems together with their theoretical background and motivation. There has been quite a number of articles which investigated the practical efficiency of the different approaches.

b-Matching Algorithms

So far we have introduced the combinatorial structures which are relevant when working with b-matchings in graphs and we have extended the concept of extreme matchings to the b-matching case. Again this concept builts the basis for the shortest augmenting path approach.

Basic Structures and Operations

Let G = (V, E) be a graph and (b v | v ∈ V) an integer vector of so-called node constraints or degree requirements. Then a mapping x: E → ℕ0 resp. vector x ∈ ℕ 0 E is called a perfect b-matching on G if