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Offline green bin packing and its constrained variant

Mingyang Gong, Brendan Mumey

Abstract

In this paper, we study the {\em green bin packing} (GBP) problem where $β\ge 0$ and $G \in [0, 1]$ are two given values as part of the input. The energy consumed by a bin is $\max \{0, β(x-G) \}$ 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 $β= 0$ 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

Offline green bin packing and its constrained variant

Abstract

In this paper, we study the {\em green bin packing} (GBP) problem where and are two given values as part of the input. The energy consumed by a bin is where 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 or , 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 . We present an APTAS and a -approximation algorithm for both GBP and CGBP, where the ratio matches the lower bound of BP. Keywords: Green bin packing; constrained green bin packing; approximation scheme; offline algorithms
Paper Structure (15 sections, 9 theorems, 15 equations)

This paper contains 15 sections, 9 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Online BP and GBP
  3. Offline GBP and its constrained variant
  4. Our contributions
  5. Preliminaries
  6. An APTAS for GBP and CGBP when $G \ge \frac{\epsilon}{3}$
  7. Handling large items
  8. Handling medium items
  9. The complete APTAS
  10. An APTAS for GBP and CGBP when $G < \frac{\epsilon}{3}$
  11. $b_\ell(\sigma^*) \ne 0$
  12. $b_\ell(\sigma^*) = 0$ and $S({\cal {T}}) \ge 2G (b_h(\sigma_1) - |{\cal {M}}|)$
  13. $b_\ell(\sigma^*) = 0$ and $S({\cal {T}}) < 2G (b_h(\sigma_1) - |{\cal {M}}|)$
  14. A $\frac{3}{2}$-approximation for GBP and CGBP
  15. Conclusions

Key Result

Lemma 1

There exists a packing $\sigma_1$ for ${\cal {L}}' \cup {\cal {M}} \cup {\cal {T}}$ such that

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • Definition 4
  • Lemma 3
  • Theorem 1
  • Lemma 4
  • Lemma 5
  • ...and 4 more