Fully dynamic bin packing revisited

Abstract

We consider the fully dynamic bin packing problem, where items arrive and depart in an online fashion and repacking of previously packed items is allowed. The goal is, of course, to minimize both the number of bins used as well as the amount of repacking. A recently introduced way of measuring the repacking costs at each timestep is the migration factor, defined as the total size of repacked items divided by the size of an arriving or departing item. Concerning the trade-off between number of bins and migration factor, if we wish to achieve an asymptotic competitive ration of 1 + ǫ for the number of bins, a relatively simple argument proves a lower bound of Ω( 1 /ǫ) for the migration factor. We establish a nearly matching upper bound of O( 1 /ǫ 4 log 1 /ǫ) using a new dynamic rounding technique and new ideas to handle small items in a dynamic setting such that no amortization is needed. The running time of our algorithm is polynomial in the number of items n and in 1 /ǫ. The previous best trade-off was for an asymptotic competitive ratio of 5 /4 for the bins (rather than 1 + ǫ) and needed an amortized number of O(log n) repackings (while in our scheme the number of repackings is independent of n and non-amortized). * Supported by DFG Project, Entwicklung und Analyse von effizienten polynomiellen Approximationsschemata für Scheduling-und verwandte Optimierungsprobleme, Ja 612/14-1. Algorithm: Insertionif SIZE(I(t)) < (m + 2)( 1 /δ + 2) or SIZE(I(t)) < 8( 1 /δ + 1) then use offline Bin Packing else improve(2); insert(i); // Shifting to the correct interval Let J i be the interval containing ∆(t); Let J j be the interval containingRename (A, B); improve(1); shiftA;Algorithm: Deletion if SIZE(I(t)) < (m + 2)( 1 /δ + 2) or SIZE(I(t)) < 8( 1 /δ + 1) then use offline Bin Packing else // Departing item i improve(4); delete(i); ReduceComponents; // // Shifting to the correct interval Let J i be the interval containing ∆(t); Let J j be the interval containingA(t)+B(t) ; Set d = i − j; if k(t) < k(t − 1) then // Modulo A(t) + B(t) when k decreases d = d -(A(t)+B(t)); // Shifting d groups from A to B for p := 0 to |d| − 1 do if i-p = 0 then Rename(A,B); improve(3); shiftB;

Robust Online Algorithms for Certain Dynamic Packing Problems

“…In contrast, Feldkord et al [14] presented a simple robust 1 + ǫ-competitive algorithm with amortized migration factor that also works for the dynamic case. Berndt et al also showed that a worst-case migration factor of Ω(1/ǫ) is needed [5]. This lower bound was shown to also hold for amortized migration by Feldkord et al [14].…”

“…Jansen and Klein [17] designed new techniques in order to obtain a migration factor polynomial in 1/ǫ. These techniques were improved by Berndt et al [5] to also handle the dynamic version of the Bin Packing problem where items can also depart. In all of these works, the migration factor is worst-case and can thus not be saved up for later use.…”

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

“…The development of advanced LP/ILP-techniques made this notable improvement possible. Furthermore, in [4] Berndt, Jansen, and Klein built upon the techniques developed in [18] to give an AFPTAS for fully dynamic bin packing with a similar migration factor.…”

“…A general approach in the design of robust online algorithms is to rethink existing algorithmic strategies that work for the corresponding offline problem in a way that the algorithm can adapt to a changing problem instance. The experiences that were made so far in the design of robust algorithms (see [18,4,26]) are to design the algorithm in a way such that the generated solutions fulfill very tight structural properties. Such solutions can then be adapted more easily as new items arrive.…”

“…The development of advanced LP/ILP-techniques made this notable improvement possible. Furthermore, in [4] Berndt, Jansen, and Klein built upon the techniques developed in [18] to give an AFPTAS for fully dynamic bin packing with a similar migration factor.Recently, Gupta et al[12] studied fully dynamic bin packing with several repacking measures. In the amortized migration model, they presented an APTAS with amortized migration factor O (1/ ).…”