Packing problems

About: Packing problems is a research topic. Over the lifetime, 2038 publications have been published within this topic receiving 43788 citations.

A phase diagram for jammed matter

Abstract: The search for the most efficient way of filling a container with balls is one of the oldest of mathematical puzzles. Aside from its intrinsic interest, the problem has practical relevance in systems as varied as granular processing, fruit packing, colloid behaviour and in living cells. Experiments have shown that the loosest way to pack spheres (random loose packing) gives a density of about 55% and that the most compact (random close packing, or RCP) gives a maximum density of about 64%. These values appear robust, but there is as yet no physical interpretation for them. Now Chaoming Song et al. show analytically that, indeed, spheres cannot pack in three dimensions above the 63.4% limit found by experiment. The limit arises from a statistical picture of the jammed states in which the RCP can be defined as the ground state of the ensemble of jammed matter. These results ultimately lead to a phase diagram for jammed matter that provides a unifying view of the sphere-packing problem. This paper presents a statistical description of jammed states in which random close packing can be interpreted as the ground state of the ensemble of jammed matter. The approach demonstrates that random packings of hard spheres in three dimensions cannot exceed a limit of ∼63.4 per cent. A phase diagram provides a common view of the hard sphere packing problem and illuminates various data, including the random loose packing state. The problem of finding the most efficient way to pack spheres has a long history, dating back to the crystalline arrays conjectured1 by Kepler and the random geometries explored2 by Bernal. Apart from its mathematical interest, the problem has practical relevance3 in a wide range of fields, from granular processing to fruit packing. There are currently numerous experiments showing that the loosest way to pack spheres (random loose packing) gives a density of ∼55 per cent4,5,6. On the other hand, the most compact way to pack spheres (random close packing) results in a maximum density of ∼64 per cent2,4,6. Although these values seem to be robust, there is as yet no physical interpretation for them. Here we present a statistical description of jammed states7 in which random close packing can be interpreted as the ground state of the ensemble of jammed matter. Our approach demonstrates that random packings of hard spheres in three dimensions cannot exceed a density limit of ∼63.4 per cent. We construct a phase diagram that provides a unified view of the hard-sphere packing problem and illuminates various data, including the random-loose-packed state.

Approximation schemes for covering and packing problems in image processing and VLSI

Abstract: A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NP-complete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error e > O, with running time that is polynomial when e is fixed. Though the polynomiality of these algorithms depends on the degree of approximation e being fixed, they cannot be improved, owing to a negative result stating that there are no fully polynomial approximation schemes for strongly NP-complete problems unless NP = P.The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems. The method of using the technique and how it varies with problem parameters are illustrated. A similar technique, independently devised by B. S. Baker, was shown to be applicable for covering and packing problems on planar graphs.