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A Framework for Efficient Approximation Schemes on Geometric Packing Problems of $d$-dimensional Fat Objects

Vítor Gomes Chagas, Elisa Dell'Arriva, Flávio Keidi Miyazawa

TL;DR

This work presents a general framework for efficient approximation schemes in geometric packing with $d$-dimensional convex fat objects, encapsulating problems such as multiple knapsack, bin packing, and strip packing within a unified IP-based approach. The core ideas hinge on a gap-structured partition that separates medium items from level items, a generalized $d$-dimensional Next Fit Decreasing Height packing, and a configuration-based linear program whose fractional solution is balanced and rounded to yield super-optimal packings in augmented bins. The framework yields ERAS and $ extrm{EPTAS}^+$ results for a broad class of problems (including CKP, CMKP, CBP, CMMSB, CMSP) and extends naturally to fat objects and features like item multiplicity and rotation, providing a versatile toolbox for geometric packing with strong theoretical guarantees. By combining decomposition, algebraic feasibility checks, and careful rounding, the method achieves scalable, provable approximations across diverse problem variants with practical augmentation bounds and, in many cases, PTAS or EPTAS behaviors.

Abstract

We investigate approximation algorithms for several fundamental optimization problems on geometric packing. The geometric objects considered are very generic, namely $d$-dimensional convex fat objects. Our main contribution is a versatile framework for designing efficient approximation schemes for classic geometric packing problems. The framework effectively addresses problems such as the multiple knapsack, bin packing, multiple strip packing, and multiple minimum container problems. Furthermore, the framework handles additional problem features, including item multiplicity, item rotation, and additional constraints on the items commonly encountered in packing contexts. The core of our framework lies in formulating the problems as integer programs with a nearly decomposable structure. This approach enables us to obtain well-behaved fractional solutions, which can then be efficiently rounded. By modeling the problems in this manner, our framework offers significant flexibility, allowing it to address a wide range of problems and incorporate additional features. To the best of our knowledge, prior to this work, the known results on approximation algorithms for packing problems were either highly fixed for one problem or restricted to one class of objects, mainly polygons and hypercubes. In this sense, our framework is the first result with a general toolbox flavor in the context of approximation algorithms for geometric packing problems. Thus, we believe that our technique is of independent interest, being possible to inspire further work on geometric packing.

A Framework for Efficient Approximation Schemes on Geometric Packing Problems of $d$-dimensional Fat Objects

TL;DR

This work presents a general framework for efficient approximation schemes in geometric packing with -dimensional convex fat objects, encapsulating problems such as multiple knapsack, bin packing, and strip packing within a unified IP-based approach. The core ideas hinge on a gap-structured partition that separates medium items from level items, a generalized -dimensional Next Fit Decreasing Height packing, and a configuration-based linear program whose fractional solution is balanced and rounded to yield super-optimal packings in augmented bins. The framework yields ERAS and results for a broad class of problems (including CKP, CMKP, CBP, CMMSB, CMSP) and extends naturally to fat objects and features like item multiplicity and rotation, providing a versatile toolbox for geometric packing with strong theoretical guarantees. By combining decomposition, algebraic feasibility checks, and careful rounding, the method achieves scalable, provable approximations across diverse problem variants with practical augmentation bounds and, in many cases, PTAS or EPTAS behaviors.

Abstract

We investigate approximation algorithms for several fundamental optimization problems on geometric packing. The geometric objects considered are very generic, namely -dimensional convex fat objects. Our main contribution is a versatile framework for designing efficient approximation schemes for classic geometric packing problems. The framework effectively addresses problems such as the multiple knapsack, bin packing, multiple strip packing, and multiple minimum container problems. Furthermore, the framework handles additional problem features, including item multiplicity, item rotation, and additional constraints on the items commonly encountered in packing contexts. The core of our framework lies in formulating the problems as integer programs with a nearly decomposable structure. This approach enables us to obtain well-behaved fractional solutions, which can then be efficiently rounded. By modeling the problems in this manner, our framework offers significant flexibility, allowing it to address a wide range of problems and incorporate additional features. To the best of our knowledge, prior to this work, the known results on approximation algorithms for packing problems were either highly fixed for one problem or restricted to one class of objects, mainly polygons and hypercubes. In this sense, our framework is the first result with a general toolbox flavor in the context of approximation algorithms for geometric packing problems. Thus, we believe that our technique is of independent interest, being possible to inspire further work on geometric packing.
Paper Structure (29 sections, 60 theorems, 14 equations, 4 figures, 1 table, 4 algorithms)

This paper contains 29 sections, 60 theorems, 14 equations, 4 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Our technique
  4. Other related work
  5. Organization of the text
  6. Preliminaries
  7. Circle Packing and Gap-Structured Partition
  8. The Next Fit Decreasing Height Algorithm
  9. The Circle Multiple Knapsack Problem
  10. Structural Theorem for the CKP
  11. Packing of the Medium Items
  12. Structured Packing of the Level Items
  13. Efficient Resource Augmentation Scheme for CKP and CMKP
  14. Resource Augmentation in only One Dimension
  15. PTAS for the CKP and $\textrm{EPTAS}^+$ for the CMKP
  16. ...and 14 more sections

Key Result

Lemma 1

Given a set $\mathcal{I}$ of circles, with $\lvert\mathcal{I}\rvert = n$, and an $\varepsilon h$-packing of $\mathcal{I}$ into a bin $B_{w \times h}$, for some $\varepsilon> 0$, we can find a packing of $\mathcal{I}$ into a bin of size $w \times (1 + n\sqrt{6\varepsilon})h$ in linear time.

Figures (4)

  • Figure 1: Sketch to illustrate the partitioning of the original instance.
  • Figure 2: Illustration of a structured packing of the level items.
  • Figure 3: $\mathrm{NFDH}$ applied for a set of squares.
  • Figure 6: Transforming the packing of medium items in a bin of size $(1 + \varepsilon) w \times 4\varepsilon h$ obtained by $\mathrm{NFDH}$ into a packing in a bin of size $w \times (1 + 8\varepsilon h)$.

Theorems & Definitions (61)

  • Lemma 1: MiyazawaEtal2015a
  • Lemma 2: MiyazawaEtal2015a
  • Definition 3
  • Lemma 3: MiyazawaEtal2015a
  • Lemma 4: MiyazawaEtal2015a
  • Theorem 5: Meir1968
  • Corollary 6
  • Theorem 7: Harren2009
  • Corollary 7
  • Theorem 8
  • ...and 51 more