Offline black and white bin packing

Abstract

We define and study a variant of bin packing called unrestricted black and white bin packing.Similarly to standard bin packing, a set of items of sizes in [0, 1] are to be partitioned into subsets of total size at most 1, called bins. Items are of two types, called black and white, and the item types must alternate in each bin, that is, two items of the same type cannot be assigned consecutively into a bin. Thus, a subset of items of total size at most 1 can form a valid bin if and only if the absolute value of the difference between the numbers of black items and white items in the subset is at most 1. We study this problem both with respect to the absolute and the asymptotic approximation ratios. We design a fast heuristic whose absolute approximation ratio is 2. We also design an APTAS and modify it into an AFPTAS. The APTAS can be used as an algorithm of absolute approximation ratio 3 2 , which is the best possible absolute approximation ratio for the problem unless P = NP.

On Colorful Bin Packing Games

“…By Corollary 4.5, B τ can only store items of type (1), (3) and (4). The load coming from items of type (1) packed into B τ can be at most 1 − 2kδ (since at most k items of type (1) can be packed into the same bin), so that bin B τ can potentially store all items of type (3). This implies that all of these items must indeed be packed into B τ , otherwise the player owning any of the leftover ones would lower her cost by migrating to B τ .…”

“…Hence, we can conclude that B τ stores exactly 2k white items and the item on top of B τ is black. This implies that B τ has to store at least 2k black items none of which belonging to type (2), that is, B τ has to store all items of type (3).…”

“…If the item on top of B τ is black, we get a contradiction as the player owning an item of type (1) not packed into B τ would lower her cost by migrating to B τ . If the item on top of B τ is white, it follows that B τ can store at most k − 1 items of type (3). Given that B τ contains at most k items of type (3), it follows that there is an item of this type, say x j which is packed neither into B τ nor into B τ .…”

“…The offline version of the black and white bin packing problem was considered in [3]. Most of the literature on colorful bin packing is about the online version of the problem.…”

Modelos Pseudo Polinomiais para o Problema do Empacotamento Colorido

“…Este problema foi proposto por Balogh et al (2013), que consideravam uma variante com apenas duas cores. Balogh et al (2015) apresentaram um AFPTAS e um APTAS, que por sua vez apresenta uma razão de aproximação absoluta de 3/2, para esta variante. Posteriormente, Böhm et al (2018) propuseram análises de vários algoritmos online para a variante com mais de duas cores, incluindo um algoritmo 3/2-competitivo.…”