New Algorithms for Bin Packing

In the bin-packing problem a list <italic>L</italic> of <italic>n</italic> numbers are to be packed into unit-capacity bins. For any algorithm <italic>S</italic>, let <italic>r</italic>(<italic>S</italic>) be the maximum ratio <italic>S</italic>(<italic>L</italic>)/<italic>L</italic><supscrpt>*</supscrpt> for large <italic>L</italic><supscrpt>*</supscrpt>, where <italic>S</italic>(<italic>L</italic>) denotes the number of bins used by <italic>S</italic> and <italic>L</italic><supscrpt>*</supscrpt> denotes the minimum number needed. An on-line <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>)-time algorithm RFF with <italic>r</italic>(RFF) = 5/3 and an off-line polynomial-time algorithm RFFD with <italic>r</italic>(RFFD) ≤ 11/9 - ε for some fixed ε > 0, are given. These are strictly better, respectively, than two prominent algorithms: the First-Fit (FF), which is on-line with <italic>r</italic>(FF) = 17/10, and the First-Fit-Decreasing (FFD) with <italic>r</italic>(FFD) = 11/9. Furthermore, it is shown that any on-line algorithm <italic>S</italic> must have <italic>r</italic>(<italic>S</italic>) ≥ 3/2. The question, “How well can an <italic>&ogr;</italic>(<italic>n</italic> log <italic>n</italic>)-time algorithm perform?” is also discussed. It is shown that in the generalized <italic>d</italic>-dimensional bin packing, any <italic>&ogr;</italic>(<italic>n</italic> log <italic>n</italic>)-time algorithm <italic>S</italic> must have <italic>r</italic>(<italic>S</italic>) ≥ <italic>d</italic>.