By Brian Osserman
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Additional resources for An algebra lemma
The main remaining open problem is to ﬁnd practical algorithms with better approximation factors. For example we conjecture that First Fit Decreasing is actually a 4/3-approximation. Without the stacking constraint, this follows from Graham’s 4/3-approximation bounds for multiprocessor scheduling , but with the stacking constraint the best bound we can prove is 3/2. Also, can we ﬁnd a 3 4 2 -approximation (or even 3 -approximation) that slices every rectangle at most once? Finally, is there a simple PTAS for strip packing with slicing (with or without the stacking constraint)?
The 2SP problem is very well-studied , and generalizes the bin packing problem, which is equivalent to the case in which all rectangles have unit height. The current best approximation algorithm for 2SP has an approximation factor of 5/3 + ε for any ε > 0 , and was achieved after a long sequence of successive improvements [1,16,17,19]. , the height achieved is at most 5/3 + ε times the optimal height. Many other authors have proposed algorithms with asymptotic performance guarantees [4,13,11] where an additive term is allowed.
Repeat with the remainder of ri , and continue until all rectangles have been processed. In the case of 2SP-SSC, the stacking constraint must be respected when placing slices. See Figure 1. H 6 r3 5 4 3 r2 r3 2 F r3 1 r1 0 0 1 2 3 4 Fig. 1. An execution of the First Fit algorithm on a 2SP-SSC instance. Note that r3 is sliced twice, and a smaller height would be achieved without the stacking constraint. It is not hard to show that after placing each rectangle, the diﬀerence between the maximum height H and the ﬂoor F (the maximum height to which the entire strip is ﬁlled) is at most hmax .
An algebra lemma by Brian Osserman