Open-end bin packing: new and old analysis approaches

TL;DR

This paper introduces and analyzes the concept of the price of clustering in open-end bin packing problems, comparing solutions with and without item-type restrictions, and explores different variants and their properties.

Contribution

It presents a new analytical framework for open-end bin packing, including the max-OEBP and min-OEBP variants, and examines their behavior under various item size conditions.

Findings

Analysis of max-OEBP with general item sizes

Impact of small items on max-OEBP

Brief discussion of min-OEBP as a distinct problem

Abstract

We analyze a recently introduced concept, called the price of clustering, for variants of bin packing called open-end bin packing problems (OEBP). Input items have sizes, and they also belong to a certain number of types. The new concept deals with the comparison of optimal solutions for the cases where items of distinct types can and cannot be packed together, respectively. The problem is related to greedy bin packing algorithms and to batched bin packing, and we discuss some of those concepts as well. We analyze max-OEBP, where a packed bin is valid if by excluding its largest item, the total size of items is below 1. For this variant, we study the case of general item sizes, and the parametric case with bounded item sizes, which shows the effect of small items. Finally, we briefly discuss min-OEBP, where a bin is valid if the total size of its items excluding the smallest item is below 1, which is known to be an entirely different problem.