# On bin packing with clustering and bin packing with delays

## TL;DR

This paper studies two bin packing problems: one with clustering constraints on item colors and one with delays in online packing, providing improved bounds and algorithms for both problems.

## Contribution

It offers tighter bounds on the price of clustering and improves the competitive ratio for online bin packing with delays.

## Key findings

- Price of clustering bounded between 1.93558 and 1.93667.
- Improved competitive ratio for online bin packing with delays to below 3.1551.
- Simplified analysis and algorithms for both problems.

## Abstract

We continue the study of two recently introduced bin packing type problems, called bin packing with clustering, and online bin packing with delays. A bin packing input consists of items of sizes not larger than 1, and the goal is to partition or pack them into bins, where the total size of items of every valid bin cannot exceed 1. In bin packing with clustering, items also have colors associated with them. A globally optimal solution can combine items of different colors in bins, while a clustered solution can only pack monochromatic bins. The goal is to compare a globally optimal solution to an optimal clustered solution, under certain constraints on the coloring provided with the input. We show close bounds on the worst-case ratio between these two costs, called "the price of clustering", improving and simplifying previous results. Specifically, we show that the price of clustering does not exceed 1.93667, improving over the previous upper bound of 1.951, and that it is at least 1.93558, improving over the previous lower bound of 1.93344. In online bin packing with delays, items are presented over time. Items may wait to be packed, and an algorithm can create a new bin at any time, packing a subset of already existing unpacked items into it, under the condition that the bin is valid. A created bin cannot be used again in the future, and all items have to be packed into bins eventually. The objective is to minimize the number of used bins plus the sum of waiting costs of all items, called delays. We build on previous work and modify a simple phase-based algorithm. We combine the modification with a careful analysis to improve the previously known competitive ratio from 3.951 to below 3.1551.