APTAS for bin packing with general cost structures

TL;DR

This paper introduces an asymptotic polynomial-time approximation scheme (APTAS) for a generalized bin packing problem where the cost depends on the number of items per bin, extending classical approaches to more complex cost functions.

Contribution

The paper develops an APTAS for a strongly NP-hard bin packing variant with general cost functions and classifies the problem's complexity based on the cost function choice.

Findings

Established an APTAS for the generalized bin packing problem

Provided a complete complexity classification based on the cost function

Extended bin packing solutions to more complex, realistic cost models

Abstract

We consider the following generalization of the bin packing problem. We are given a set of items each of which is associated with a rational size in the interval [0,1], and a monotone non-decreasing non-negative cost function f defined over the cardinalities of the subsets of items. A feasible solution is a partition of the set of items into bins subject to the constraint that the total size of items in every bin is at most 1. Unlike bin packing, the goal function is to minimize the total cost of the bins where the cost of a bin is the value of f applied on the cardinality of the subset of items packed into the bin. We present an APTAS for this strongly NP-hard problem. We also provide a complete complexity classification of the problem with respect to the choice of f.