Fair versus Unrestricted Bin Packing

The Relative Worst Order Ratio for On-Line Algorithms

“…Thus, OPT accepts all (15 · 2 q − 9) + (9 · 2 q − 6) = 24 · 2 q − 15 items. This gives a ratio of Theorem 4.10 considers special values of n. As in [Azar et al 2002], it can be shown that, in general, First-Fit n 's competitive ratio on accommodating sequences is …”

“…Unfair-First-Fit [Azar et al 2002] is a variant of First-Fit designed to do better than First-Fit on certain sequences containing many large items followed by many small items. It has a better competitive ratio on accommodating sequences than First-Fit, but on sequences containing only large items it actually does worse than First-Fit.…”

“…In contrast, in the Dual Bin Packing problem, we are given a fixed number n of unit sized bins, and the goal is to maximize the number of items packed in the n bins. A variant of Dual Bin Packing is the Fair Bin Packing problem Azar et al 2002], where the algorithms have to be fair, i.e., to reject items only when they do not fit in any bin.…”

“…An example of an algorithm for the Dual Bin Packing problem which is not fair is Unfair-FirstFit [Azar et al 2002]. It behaves as First-Fit, except that when given an item of size greater than 1 2 , it automatically rejects that item if it has already accepted at least 2 3 of the items seen so far.…”

“…To show that First-Fit n has the same competitive ratio on accommodating sequences as First-Fit asymptotically, we use the sequences from [Azar et al 2002] showing that First-Fit's competitive ratio on accommodating sequences is 9 · 2 q − 6 items of size E = 1 3 . In each phase j, 1 ≤ j ≤ q, First-Fit n will pair each item of size B j with one item of size C j , thus using 3 · (2 q+1 − 2) + 1 = 6 · 2 q − 5 bins for these q phases and the phase 0.…”