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NP-Complete operations research problems and approximation algorithms

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Abstract

For a large number of discrete optimization problems like the traveling salesman problem, the quadratic assignment problem, the general flow-shop problem, the knapsack problem etc. all known algorithms have the discouraging property that their (worst-case) running times on a computer grow exponentially with the size of the problem. All efforts to find polynomial bounded algorithms for these problems have failed. Recent results in complexity theory show that these problems belong to the classes ofNP-complete orNP-hard problems. It is a common belief that for problems belonging to these classes no polynomial bounded algorithms exist. Heuristics or approximation algorithms should be applied to these problems.

The aim of this tutorial paper is to give a survey onNP-complete andNP-hard problems and on approximation algorithms. All concepts introduced are illustrated by examples which are closely related to the knapsack problem and can be understood easily. References to most other problems of interest to operations researchers are given.

Zusammenfassung

Für eine große Anzahl von diskreten Optimierungsproblemen wie das Traveling Salesman Problem, das quadratische Zuordnungsproblem, das allgemeine Flowshop Problem, das Rucksackproblem usw. haben alle bisherigen Lösungsansätze die unangenehme Eigenschaft, daß der Rechenumfang der entsprechenden Algorithmen exponentiell mit dem Umfang der Probleme wächst. Alle Bemühungen polynomial beschränkte Verfahren für solche Probleme zu finden waren bislang ergebnislos. Neuere Ergebnisse der Komplexitätstheorie besagen, daß diese Probleme zu den Klassen derNP-vollständigen bzw.NP-schwierigen Probleme gehören und somit nach allgemein verbreiteter Auffassung wohl niemals polynomial lösbar sein werden. Die Anwendung von heuristischen Verfahren oder approximativen Algorithmen scheint der einzige Ausweg in dieser Situation.

Ziel der Arbeit ist es, einen einführenden Uberblick über die Theorie derNP-vollständigen undNP-schwierigen Probleme sowie über approximative Verfahren zu geben. Alle in der Arbeit einge-führten Begriffe werden an einfach verständlichen Beispielen erläutert, die eng mit dem Rucksack-problem verwandt sind.

Für weitere Probleme, die den Unternehmensforscher interessieren, findet der Leser ausführliche Literaturübersichten.

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  1. Fachbereich IV der Universität Oldenburg, Postfach 2503, D-Oldenburg

    Peter Brucker

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An invited survey.

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Brucker, P. NP-Complete operations research problems and approximation algorithms. Zeitschrift für Operations Research 23, 73–94 (1979). https://doi.org/10.1007/BF01951543

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