Online Results for Black and White Bin Packing

Abstract

In online bin packing problems, items of sizes in [0,1] are to be partitioned into subsets of total size at most 1, called bins. We introduce a new variant where 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. We design an online algorithm with the absolute competitive ratio 3. We further show that a number of well-known algorithms cannot have a better performance, even in the asymptotic sense. Additionally, we prove a surprising general lower bound 1 + 1 2 ln 2 ≈ 1.7213 on the asymptotic competitive ratio of any deterministic or randomized online algorithm. This lower bound significantly exceeds the known upper bound 1.58889 for classic online bin packing.A proceedings version where some of the results in this article were announced appeared as [1].

Offline black and white bin packing

“…Let t be the number of huge white items, i.e., w t > 1 2 , and if n w > t, then w t+1 ≤ 1 2 (in particular, if no such item exists, t = 0). Let 0 ≤ M ≤ t be the minimum value such that for all 1 …”

“…Specifically, if i is black, we let c i = 1, and if i is white, then we let c i = −1. In papers [2,1] the variant that we study here is called the unrestricted offline variant, i.e., the input is given as a set (rather than a sequence as in the online problem). These papers describe applications of the problem, and provide motivation for its study.…”

“…A polynomial time approximation scheme (PTAS) is a family of algorithms that for every ε > 0 contains an algorithm whose absolute approximation ratio is at most 1 + ε. The running time must be polynomial in n, but not necessarily polynomial in 1 ε . A fully polynomial time approximation scheme (FPTAS) is a PTAS where the running time is polynomial also in 1 ε .…”

“…The running time must be polynomial in n, but not necessarily polynomial in 1 ε . A fully polynomial time approximation scheme (FPTAS) is a PTAS where the running time is polynomial also in 1 ε . If the approximation ratio is defined in the asymptotic sense, such schemes are called APTAS's and AFPTAS's, respectively.…”

“…Thus, our results are the best possible (unless P = NP). The extended abstract [2] contained (in addition to results for the online variant, which were expanded into paper [1]) the APTAS and a fast heuristic of absolute approximation ratio 5 2 . Standard bin packing is usually studied with respect to the asymptotic approximation ratio, but for the algorithms Next Fit [21] and First Fit [22,28] (defined in the body of the paper) it is known that their asymptotic approximation ratios, 2 and 17 10 , respectively, are actually also their absolute approximation ratios [21,11], and First Fit Decreasing (First Fit applied to a non-increasingly sorted input) has an absolute approximation ratio of 3 2 [25], which is the best possible unless P = NP.…”