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Extra resources for Approximation and Online Algorithms: 12th International Workshop, WAOA 2014, Wrocław, Poland, September 11-12, 2014, Revised Selected Papers
I) For every block Bi ⊆ D we have w(Bi ) ≥ 0. (ii) If there is no special bin in D, then w(D) ≥ 0. If also C ∈ R then w(D) ≥ 1. We are now ready to derive the ﬁnal contradiction. If D does not contain a special bin, we add the appropriate bounds from Lemmata 7, 8 and 6. If C ∈ R then F is not in the ﬁrst block and w(L) = w(C)+w(D)+w(F) > −6+1+5 = 0. If C ∈ E then F is not in the ﬁrst block and w(L) = w(C)+w(D)+w(F) > −5+0+5 = 0. If C is the last special bin then w(L) = w(C) + w(D) + w(F) > −4 + 0 + 4 = 0.
Be+1 . Thus the list L is (some or all of the ﬁrst six bins may not exist): L, M, T, R1,1 , R1,2 , R1,3 , E1 , R2,1 , R2,2 , R2,3 , E2 , . . , Ee , Re+1,1 , Re+1,2 , Re+1,3 . Whenever we refer to the ordering of the bins, we mean the ordering in L. See Fig. 1 for an illustration. We use First Fit on the reversed list L for huge items (that is, we pack each huge item to the last bin in L where it ﬁts) and we use First Fit on L for all other items. Suppose that we have an instance that has a packing into bins of capacity 12 and on which our algorithm fails.
7-competitive for all K ≥ 2 . 691 for K large enough. Lee and Lee also proved that there is no bounded space algorithm with a better asymptotic ratio. The Bounded Space Bin Packing is an especially interesting variant in our context due to the fact that it matters whether we allow the optimum to reorder the input instance or not. If we allow reordering for Bounded Space Bin Packing, we get the same optimum as classical Bin Packing. In fact, all the bounds on online algorithms in the previous paragraph hold if the optimum with reordering is considered, which is a stronger statement than comparing to the optimum without reordering.