A Robust AFPTAS for Online Bin Packing with Polynomial Migration

Abstract

We consider the relaxed online strip packing problem: Rectangular items arrive online and have to be packed without rotations into a strip of fixed width such that the packing height is minimized. Thereby, repacking of previously packed items is allowed. The amount of repacking is measured by the migration factor, defined as the total size of repacked items divided by the size of the arriving item. First, we show that no algorithm with constant migration factor can produce solutions with asymptotic ratio better than 4/3. Against this background, we allow amortized migration, i. e. to save migration for a later time step. As a main result, we present an AFPTAS with asymptotic ratio 1 + O ( ) for any > 0 and amortized migration factor polynomial in 1/ . To our best knowledge, this is the first algorithm for online strip packing considered in a repacking model. PreliminariesSince strip packing is NP-hard [1], research focuses on efficient approximation algorithms. Let A(I) denote the packing height of algorithm A on input I and OPT(I) the minimum packing height. The absolute (approximation) ratio is defined as sup I A(I)/ OPT(I) while the asymptotic (approximation) ratio as lim sup OPT(I)→∞ A(I)/ OPT(I). A family of algorithms {A } >0 is called polynomial-time approximation scheme (PTAS), when A runs in polynomial-time in the input length and has absolute ratio 1 + . If the running time is also polynomial in 1/ , we call {A } >0 fully polynomial-time approximation scheme (FPTAS). Similarly, the terms APTAS and AFPTAS are defined using the asymptotic ratio.All ratios of online algorithms in the following are competitive, i. e. online algorithms are compared with an optimal offline algorithm. Related WorkOffline Strip packing is one of the classical packing problems and receives ongoing research interest in the field of combinatorial optimization. Since Baker, Coffman and Rivest [1] gave the first algorithm with asymptotic ratio 3, strip packing was investigated in many studies, considering both asymptotic and absolute approximation ratios. We refer the reader to [6] for a survey. A well-known result is the AFPTAS by Kenyon and Rémila [21]. Concerning the absolute ratio, currently the best known algorithm of ratio 5/3 + for any > 0 is by Harren et al. [14].An interesting result was given by Han et al. in 2007. In [13], they studied the relation between bin packing and strip packing and developed a framework between both problems. For the offline case it is shown that any bin packing algorithm can be applied to strip packing while maintaining the same asymptotic ratio.Online The first algorithm for online strip packing was given by Baker and Schwarz [2] in 1983. Using the concept of shelf algorithms [1], they derived the algorithm First-Fit-Shelf with asymptotic ratio arbitrary close to 1.7 and absolute ratio 6.99 (where all rectangles have height at most h max = 1). Later, Csirik and Woeginger [8] showed a lower bound of h ∞ ≈ 1.69 on the asymptotic ratio of shelf algorithms and gave an improved shelf algorithm with a...

Fully dynamic bin packing revisited

“…The running time is polynomial in n and 1 /ǫ. In case that no deletions are used, the algorithm has a migration factor of O( 1 /ǫ 3 · log( 1 /ǫ)), which beats the best known migration factor of O( 1 /ǫ 4 ) by Jansen and Klein [JK13]. Since the number of repacked bins is bounded, so is the number of shifting moves as it requires at most O( 1 /ǫ) shifting moves to repack a single bin.…”

“…They also proved that there is no online algorithm for this problem that has a constant migration factor and that maintains an optimal solution. The APTAS by Epstein and Levin was later improved by Jansen and Klein [JK13], who developed a robust AFPTAS for the problem with migration factor O( 1 /ǫ 4 ). In their paper, they developed new linear program (LP)/integer linear program (ILP) techniques, which we make use of to obtain polynomial migration.…”

“…Epstein and Levin [EL09] used the measure of the migration factor to give an algorithm that has an asymptotic competitive ratio of 1 + ǫ and a migration factor of 2 O((1/ǫ) log 2 (1/ǫ)) . This result was improved by Jansen and Klein [JK13] who achieved polynomial migration. Their algorithm uses a migration factor of O( 1 /ǫ 4 ) to achieve an asymptotic competitive ratio of 1 + ǫ.…”

Robust Online Algorithms for Certain Dynamic Packing Problems

“…For the Bin Packing problem, Epstein and Levin [11] gave the first robust 1+ǫ-competitive algorithm for Bin Packing based on the same sensitivity results for integer programming. 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.…”

“…This problem has been studied intensively in the online setting (see for example the works cited in [8]). Jansen et al [18] studied the static case in the migration scenario, where rectangles can only arrive.…”