Offline green bin packing and its constrained variant

TL;DR

This paper introduces the green bin packing problem, incorporating energy consumption into bin packing, and provides approximation algorithms and schemes for both the unconstrained and constrained variants.

Contribution

It formulates the green bin packing and constrained variants, and develops an APTAS and a 3/2-approximation algorithm for both, matching the lower bound of classical bin packing.

Findings

Developed an APTAS for GBP and CGBP.

Designed a 3/2-approximation algorithm for GBP and CGBP.

Achieved approximation ratios that match the classical bin packing lower bound.

Abstract

In this paper, we study the {\em green bin packing} (GBP) problem where $\beta \ge 0$ and $G \in [0, 1]$ are two given values as part of the input. The energy consumed by a bin is $\max \{0, \beta (x-G) \}$ where $x$ is the total size of the items packed into the bin. The GBP aims to pack all items into a set of unit-capacity bins so that the number of bins used plus the total energy consumption is minimized. When $\beta = 0$ or $G = 1$, GBP is reduced to the classic bin packing (BP) problem. In the {\em constrained green bin packing} (CGBP) problem, the objective is to minimize the number of bins used to pack all items while the total energy consumption does not exceed a given upper bound $U$. We present an APTAS and a $\frac 32$-approximation algorithm for both GBP and CGBP, where the ratio $\frac 32$ matches the lower bound of BP. Keywords: Green bin packing; constrained green bin packing; approximation scheme; offline algorithms