| Research>>> Acta Cybernetica>>> Previous Volumes>>> | Magyarul |
Two simple algorithms for bin covering
Le]Example Le]Theorem Le]Definition
J. Csirik 1, J. B. G. Frenk 2, M. Labbé 3, S. Zhang 4
Dedicated to Professor Ferenc Gécseg on the occasion of his 60th birthday
Abstract
We define two simple algorithms for the bin covering problem and give their asymptotic performance.
1. Introduction
In this chapter we consider the following version of bin packing sometimes called dual bin packing or bin covering : given a list
|
of items with size s(ai) for each item ai, and a bin capacity C,
|
pack the elements of L into a maximum number of bins so that the sum of sizes in any bin is at least C. This means, that we have to fill as many bins as possible. It is clear, that we can normalize the problem so that C is equal to 1 and s(ai) < 1 for every 1 £ i £ n. The above problem was investigated for the first time by Assmann (cf.[]) and Assmann et al. (cf.[]). In particular, they showed that the problem is NP -hard. Furthermore, they provided the first approximation algorithms and proved their worst-case performance. Some average-case analysis was also performed.
We denote by OPT(L) the optimal, i.e. the maximal number of filled bins for a list
|
|
(1) |
where A(L) denotes the number of bins filled by algorithm A applied to the list L. The performance ratio or asymptotic worst case ratio of A, denoted by RA, is now given by
|
(2) |
Clearly, RA(k) £ 1 for every k ³ 1, and hence RA £ 1.
For an equivalent definition of RA we observe that RA ³ K1 if there exist two constants K1 and K2 such that
|
(3) |
for every list L. Clearly the largest possible K1 satisfying this inequality equals RA. By this definition it is obvious that a heuristic A1 is at least as good as heuristic A2 (from a worst case point of view) if RA1 ³ RA2.
Footnotes:
1 Institute of Informatics, József Attila University, Árpád tér 2, H-6720 Szeged, Hungary
2 Erasmus University of Rotterdam, Faculty of Economics, Postbus 1738, 3000 DR Rotterdam, Netherlands
3 Université Libre de Bruxelles, Mathematiques du Triomphe, 1050 Bruxelles, Belgium
4 Erasmus University of Rotterdam, Faculty of Economics, Postbus 1738, 3000 DR Rotterdam, Netherlands