AFPTAS results for common variants of bin packing: A new method to handle the small items
Leah Epstein, Asaf Levin

TL;DR
This paper extends AFPTAS to two variants of bin packing—cardinality constraints and rejection—by developing a novel method for packing small items, improving efficiency over previous APTAS results.
Contribution
The paper introduces a new method for packing small items that enables AFPTAS for bin packing with cardinality constraints and rejection, expanding the scope of efficient approximation schemes.
Findings
AFPTAS developed for bin packing with cardinality constraints
AFPTAS developed for bin packing with rejection
Improved running times over previous APTAS results
Abstract
We consider two well-known natural variants of bin packing, and show that these packing problems admit asymptotic fully polynomial time approximation schemes (AFPTAS). In bin packing problems, a set of one-dimensional items of size at most 1 is to be assigned (packed) to subsets of sum at most 1 (bins). It has been known for a while that the most basic problem admits an AFPTAS. In this paper, we develop methods that allow to extend this result to other variants of bin packing. Specifically, the problems which we study in this paper, for which we design asymptotic fully polynomial time approximation schemes, are the following. The first problem is "Bin packing with cardinality constraints", where a parameter k is given, such that a bin may contain up to k items. The goal is to minimize the number of bins used. The second problem is "Bin packing with rejection", where every item has a…
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