New Algorithms for Bin Packing

Andrew Chi-Chih Yao

Abstract: In the bin-packing problem a list L of n numbers are to be packed into unit-capacity bins. For any algorithm S, let r(S) be the maximum ratio S(L)/L* for large L*, where S(L) denotes the number of bins used by S and L* denotes the minimum number needed. An on-line Ο(n log n)-time algorithm RFF with r(RFF) = 5/3 and an off-line polynomial-time algorithm RFFD with r(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 r(FF) = 17/10, and the First-Fit-Decreasing (FFD) with r(FFD) = 11/9. Furthermore, it is shown that any on-line algorithm S must have r(S) ≥ 3/2. The question, “How well can an ο(n log n)-time algorithm perform?” is also discussed. It is shown that in the generalized d-dimensional bin packing, any ο(n log n)-time algorithm S must have r(S) ≥ d.