Abstract. The standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of Z"' in a suitable box. However, it suffices to consider those x e Z'" which lie in a suitable ellipsoid having a much smaller volume than the box. We show in this paper that searching through that ellipsoid is in many cases much more efficient. If combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions. Several randomly constructed numerical examples illustrate the superiority of our new method over the known ones.
Abstract:In this paper a surprising probabilistic behaviour of quadratic sum assignment problems is shown. The relative difference between worst and optimal solution value tends to zero with probability tending to one as the size of the problem goes to infinity. This result suggests that for high dimensional quadratic assignment problems even very simple approximation algorithms can in practice yield good suboptimal solutions.
Zusammenfassung:In dieser Arbeit wird ein iiberraschendes asymptotisches Verhalten yon quadratischen Zuordnungsproblemen mit Summenzielfunktion aufgezeigt. Es wixd bewiesen, daft der relative Untersehied zwischen schlechtestem und bestem Zielfunktionswert mit Wahrscheinlichkeit gegen 1 nach Null strebt, wenn die Dimension der Probleme gegen Unendlich geht. Aufgrund dieses Resultates kann man erwarten, da~ schon sehr einfache Heuristiken t'fir h6her-dimensionale Probleme in der Praxis recht gute N~iherungsl6sungen liefern.
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