# Approximation schemes for the generalized extensible bin packing problem

## TL;DR

This paper introduces an efficient approximation scheme for a generalized bin packing problem where bin costs depend on bin index, improving previous methods and providing new approximation schemes.

## Contribution

It presents an EPTAS for a new generalized bin packing problem and an AFPTAS for a related variant, advancing approximation techniques.

## Key findings

- Established an EPTAS for the generalized bin packing problem.
- Developed an AFPTAS for a related bin packing variant.
- Improved previous PTAS to an EPTAS for the problem.

## Abstract

We present a new generalization of the extensible bin packing with unequal bin sizes problem. In our generalization the cost of exceeding the bin size depends on the index of the bin and not only on the amount in which the size of the bin is exceeded. This generalization does not satisfy the assumptions on the cost function that were used to present the existing polynomial time approximation scheme (PTAS) for the extensible bin packing with unequal bin sizes problem. In this work, we show the existence of an efficient PTAS (EPTAS) for this new generalization and thus in particular we improve the earlier PTAS for the extensible bin packing with unequal bin sizes problem into an EPTAS. Our new scheme is based on using the shifting technique followed by a solution of polynomial number of $n$-fold programming instances. In addition, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) for the related bin packing type variant of the problem.