Hardness of Online Translational Packing of Convex Polygons

المفاهيم الأساسية

Online algorithms for packing convex polygons by translation alone do not admit constant competitive ratios, in contrast to the case where rotations are allowed or the offline setting.

الملخص

The paper investigates several online packing problems where convex polygons arrive one by one and must be placed irrevocably into a container, with the goal of minimizing the used space. The key findings are:

Online algorithms for packing convex polygons by translation alone do not admit constant competitive ratios, in contrast to the case where rotations are allowed or the offline setting.

To prove these lower bounds, the authors introduce a novel combinatorial problem called "Online Sorting" and show that it does not admit constant competitive algorithms either. This reveals a deep connection between geometric packing problems and purely combinatorial problems.

On the positive side, the authors present non-trivial algorithms for both online sorting and online strip packing of convex polygons.

The authors show that sorting the pieces by the slope of their "spine segments" is essential for obtaining efficient offline packing algorithms, but this sorting step is inherently difficult in the online setting.

The results highlight the challenges in extending efficient offline packing algorithms to the online setting, especially when rotations are not allowed.

الإحصائيات

None.

اقتباسات

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الرؤى الأساسية المستخلصة من

Online Sorting and Translational Packing of Convex Polygons

by Anders Aaman... في arxiv.org 04-09-2024

https://arxiv.org/pdf/2112.03791.pdf

استفسارات أعمق

What are some potential applications or real-world scenarios where the hardness of online translational packing of convex polygons could have significant practical implications

The hardness of online translational packing of convex polygons can have significant practical implications in various applications. One potential application is in the field of manufacturing and production, particularly in industries where materials need to be cut or shaped efficiently. For example, in the textile industry, where fabric pieces are cut from larger rolls of fabric, optimizing the placement of patterns on the fabric to minimize waste is crucial. The ability to pack convex polygons efficiently in an online setting could lead to reduced material wastage and increased cost-effectiveness in production processes.
Another application could be in the field of logistics and transportation, where items of irregular shapes need to be packed into containers or vehicles. Efficiently packing convex polygons in an online scenario could help in maximizing the use of available space, reducing the number of trips or shipments required, and ultimately improving the overall efficiency of the logistics operations.

How might the techniques developed in this paper for proving lower bounds on the competitive ratio of online packing problems be applicable to other online optimization problems beyond packing and sorting

The techniques developed in this paper for proving lower bounds on the competitive ratio of online packing problems can be applied to other online optimization problems beyond packing and sorting. These techniques involve constructing adversarial streams of inputs to challenge the performance of online algorithms, leading to lower bounds on their competitive ratios. Similar approaches can be used in various online optimization problems such as online scheduling, online routing, online matching, and online resource allocation.
By designing adaptive and deterministic strategies to present inputs that challenge the algorithms, researchers can establish lower bounds on the competitive ratios of online algorithms in different problem domains. This can provide insights into the inherent difficulty of online optimization problems and guide the development of more efficient algorithms in various applications.

Could the online sorting problem introduced in this paper have further implications or connections to other areas of computer science beyond packing and scheduling, such as online algorithms, combinatorial optimization, or data structures

The online sorting problem introduced in this paper can have further implications and connections to other areas of computer science beyond packing and scheduling. One potential connection is with online algorithms, where the goal is to design algorithms that make decisions in an online fashion without knowledge of future inputs. The techniques developed for online sorting, such as constructing adversarial streams and proving lower bounds on competitive ratios, can be applied to analyze the performance of online algorithms in different contexts.
Furthermore, the concept of online sorting can be related to combinatorial optimization problems, where the goal is to find the best arrangement or selection of elements from a finite set. The study of online sorting in the context of convex polygons can provide insights into the complexity of arranging geometric objects in an online setting and may lead to new algorithmic approaches for optimization in combinatorial problems.
Additionally, the online sorting problem may have connections to data structures and algorithms, particularly in the design of efficient data structures for sorting and searching operations in dynamic or online environments. Understanding the challenges and limitations of online sorting can contribute to the development of data structures that perform well in scenarios with evolving or unpredictable inputs.