Beating the Harmonic lower bound for online bin packing

Sandy Heydrich; Rob van Stee

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arXiv · 1511.00876

Beating the Harmonic lower bound for online bin packing

Sandy Heydrich·Rob van Stee

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Abstract

In the online bin packing problem, items of sizes in (0,1] arrive online to be packed into bins of size 1. The goal is to minimize the number of used bins. In this paper, we present an online bin packing algorithm with asymptotic competitive ratio of 1.5813. This is the first improvement in fifteen years and reduces the gap to the lower bound by 15%. Within the well-known SuperHarmonic framework, no competitive ratio below 1.58333 can be achieved. We make two crucial changes to that framework. First, some of our algorithm's decisions depend on exact sizes of items, instead of only their types. In particular, for each item with size in (1/3,1/2], we use its exact size to determine if it can be packed together with an item of size greater than 1/2. Second, we add constraints to the linear programs considered by Seiden, in order to better lower bound the optimal solution. These extra constraints are based on marks that we give to items based on how they are packed by our algorithm. We show that for each input, a single weighting function can be constructed to upper bound the competitive ratio on it. We use this idea to simplify the analysis of SuperHarmonic, and show that the algorithm Harmonic++ is in fact 1.58880-competitive (Seiden proved 1.58889), and that 1.5884 can be achieved within the SuperHarmonic framework. Finally, we give a lower bound of 1.5762 for our new framework.

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Sandy Heydrich, Rob van Stee. 2015-11-03. Beating the Harmonic lower bound for online bin packing. https://arxiv.org/abs/1511.00876

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