a b s t r a c tIn this paper, we present improved bounds for the First Fit algorithm for the bin-packing problem. We prove C FF (L) ≤ 17 10 C * (L)+ 7 10 for all lists L, and the absolute performance ratio of FF is at most 12 7 .

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This paper considers two semi-online scheduling problems, one with known optimal value and the other with known total sum, on two uniform machines with a machine speed ratio of s ≥ 1. For the first problem, we provide an optimal algorithm for s ∈ [ 1+ √ 3 2 , 1+ √ 21 4 ], and improved algorithms or/and lower bounds for s ∈ [ 1+ √ 21 4, √ 3], over which the optimal algorithm is unknown. As a result, the largest gap between the competitive ratio and the lower bound decreases to 0.02192. For the second problem, we also present algorithms and lower bounds for s ≥ 1. The largest gap between the competitive ratio and the lower bound is 0.01762, and the length of the interval over which the optimal algorithm is unknown is 0.47382. Our algorithms and lower bounds for these two problems provide insights into their differences, which are unusual from the viewpoint of the known results on these two semi-online scheduling problems in the literature. Mathematics Subject Classification (1991). 90B35, 90C27This is the Pre-Published Version.is achievable (i.e., the optimal value) for the sequence is known in advance. We call this problem the semi-online scheduling problem with known optimal value, and denoted it by P m|opt|C max if all the machines have the same speed 1, and by Qm|opt|C max otherwise. Azar and Regev [2] gave an application of this problem in file allocation.A closely related problem is the semi-online scheduling problem with known total sum, where the total sum of all the job sizes is known in advance [7]. We denote this problem by P m|sum|C max if all the machines have the same speed 1, and by Qm|sum|C max otherwise. As the total sum of all the job sizes gives a trivial lower bound for the optimal value, the problem with known total sum may be viewed as a relaxation of the problem with known optimal value. But these two problems are clearly different, since, among other reasons, some jobs may have sizes greater than the average and hence the optimal makespan is still unknown.The quality of the performance of an on-line or a semi-online algorithm is measured by its competitive ratio. For a job sequence J and an algorithm A, let C A (J ) (or briefly C A ) denote the makespan produced by A and let C OP T (J ) (or briefly C OP T ) denote the optimal makespan of the off-line version. Then the competitive ratio of A is defined as R A = sup J √ 17 4 ≈ 1.28078, s, for 1+