AFPTAS results for common variants of bin packing: A new method to handle the small items

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 rejection penalty associated with it. An item needs to be either packed to a bin or rejected, and the goal is to minimize the number of used bins plus the total rejection penalty of unpacked items. This resolves the complexity of two important variants of the bin packing problem. Our approximation schemes use a novel method for packing the small items. This new method is the core of the improved running times of our schemes over the running times of the previous results, which are only asymptotic polynomial time approximation schemes (APTAS).