Zusammenfassung
Wir untersuchen Mengenüberdeckungsprobleme mitn Variablen undm Nebenbedingungen, in denen nicht mehr alsk Variable in jeder Restriktion auftreten. Seid die maximale Spaltensumme,r=[m/d]−1 undZ g der Wert einer Greedy-Lösung. Wir zeigen
Dies ist die erste nichttriviale oberea-priori-Schranke für Mengenüberdeckungsprobleme. Ein Beispiel zeigt, daß dieser Wert sowohl von der Greedy-Lösung wie auch von der Optimallösung angenommen werden kann. Ferner belegen numerische Beispiele, daß diese Schranke für viele spezielle Probleme wesentlich besser ist als bereits bekannte Schranken. Eine wichtige Teilklasse, für die dies zutrifft, umfaßt Probleme, deren Restriktionsmatrix eine Zirkulante undk=d=[αn] für ein 0<α<1 ist. In diesem Falle zeigen wir
Abstract
We study the class of (m constraint,n variable) set covering problems which have no more thank variables represented in each constraint. Letd denote the maximum column sum in the constraint matrix, letr=[m/d]−1, and letZ g denote the cost of a greedy heuristic solution. Then we prove
This provides the firsta priori nontrivial upper bound discovered on heuristic solution cost (and thus on optimal solution cost) for the set covering problem. An example demonstrates that this bound is attainable, both for a greedy heuristic solution and for the optimal solution. Numerical examples show that this bound is substantially better than existing bounds for many problem instances. An important subclass of these problems occurs when the constraint matrix is a circulant, in which casem=n andk=d=[αη] for some 0<α<1. For this subclass we prove
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Hall, N.G., Vohra, R.V. Absolute bounds on optimal cost for a class of set covering problems. ZOR - Methods and Models of Operations Research 33, 181–192 (1989). https://doi.org/10.1007/BF01423649
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DOI: https://doi.org/10.1007/BF01423649