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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].

We improve the lower bound on the asymptotic competitive ratio of any online algorithm for bin packing to above 1.54278. We demonstrate for the first time the advantage of branching and the applicability of full adaptivity in the design of lower bounds for the classic online bin packing problem. We apply a new method for weight based analysis, which is usually applied only in proofs of upper bounds. The values of previous lower bounds were approximately 1.5401 and 1.5403.

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.