A new framework for analyzing online bin packing algorithms is presented. This framework presents a unified way of explaining the performance of algorithms based on the Harmonic approach. Within this framework, it is shown that a new algorithm, Harmonic++, has asymptotic performance ratio at most 1.58889. It is also shown that the analysis of Harmonic+1 presented in Richey [1991] is incorrect; this is a fundamental logical flaw, not an error in calculation or an omitted case. The asymptotic performance ratio of Harmonic+1 is at least 1.59217. Thus, Harmonic++ provides the best upper bound for the online bin packing problem to date.

Abstract.A new framework for analyzing online bin packing algorithms is presented. This framework presents a unified way of explaining the performance of algorithms based on the Harmonic approach [3,5,8,10,11,12]. Within this framework, it is shown that a new algorithm, Harmonic++, has asymptotic performance ratio at most 1.58889. It is also shown that the analysis of Harmonic+1 presented in [11] is incorrect; this is a fundamental logical flaw, not an error in calculation or an omitted case. The asymptotic performance ratio of Harmonic+1 is at least 1.59217. Thus Harmonic++ provides the best upper bound for the online bin packing problem to date.

We study the problem of on-line scheduling on two uniform machines with speeds 1 and s*1. A +1.61803 competitive deterministic algorithm was already known. We present the "rst randomized results for this problem: We show that randomization does not help for speeds s*2, but does help for all s(2. We present a simple memoryless randomized algorithm with competitive ratio (4!s)(1#s)/4)1.56250. We analyse other randomized algorithms that demonstrate that the randomized competitive ratio is at most 1.52778 for any s. These algorithms are barely random, i.e. they use only a constant number of random bits. Finally, we present a 1#s/(s#s#1) competitive deterministic algorithm for the preemptive version of this problem. For any s, it is best possible even among randomized preemptive algorithms.

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