To measure the efficiency of algorithms, two general methods are available: an investigation of the worst-case behavior or, assuming some probability measures, a probabilistic analy- sis. Here we will concentrate on the asymptotic worst-case behavior of an algorithm. For a given list L, let A(L) and
OPT (L) stand for the number of bins used by algorithm A and the number of bins used in an optimal packing, re-
spectively. Then the asymptotic competitive ratio (ACR) of algorithm A is
R(A) := lim sup
l→∞ max L A(L) l OPT(L) = l. (1)
For off-line algorithms Fernandez de la Vega and Lueker [14] provided an APTAS (Asymptotic Polynomial Time Approx- imation Scheme), while Karmakar and Karp [28] developed the first AFPTAS (Asymptotic Fully Polynomial Time Ap- proximation Scheme). In [14] for any ε > 0 an algorithm Aε
is given such that each Aε runs in polynomial time in the
length of the input list L (but exponential in 1/ε) and has
Aε(L) = (1 + ε) OPT (L) + 1. In [28], a more complex algo- rithm is given. The running time of this depends on n and 1/ε polynomially and Aε(L) ≤ OPT (L) + log2(OPT (L))
holds for this.
The current best on-line algorithm with the best known ACR was defined by Seiden [34] in 2002, and it is called
Harmonic++. Seiden (improving Richey’s earlier method
and analysis [34]) proved that the ACR of his algorithm is at most 1.58889. Harmonic++ belongs to the class of Super
Harmonic algorithms defined in the same paper. For each
algorithm from this class, a lower bound of 1.58333 is valid [32], so 1.58333 ≤ R(Harmonic++) ≤ 1.58889. Note that the last previous actual best algorithms were Super Har- monic [29, 32, 34] as well. For on-line algorithms the best known lower bound is 1.5403 [1].
In this paper we will focus on semi-on-line (SOL) bin pack- ing algorithms. The most famous algorithms based on pre- ordering are FFD and BFD. They sort the elements in de- creasing order and pack them using the First Fit (Best Fit) strategy. Johnson proved in [27] that R(FFD) = R(BFD) = 11/9 = 1.22222.... More precisely, Johnson proved that for each list L FFD(L) ≤ (11/9) OPT (L) + 4. The tight value 6/9 of the additive term was given by D´osa [11], proving that FFD(L)≤ 11/9 OPT (L) + 6/9. For on-line algorithms running on pre-ordered lists we improved the previous lower bound from 8/7 = 1.1428... [9] in [1] to 54/47 = 1.1489.... The first semi-on-line bin packing algorithm with repacking was given by Galambos [15] for the case when a restricted number of bins can be open. A bin is called closed if we cannot pack elements into it later. This algorithm uses two buffer bins for storing the elements temporarily. Improving on this, Galambos and Woeginger [17] defined a semi-on- line repacking algorithm using 3 buffer bins. Its ACR is 1.69103..., which is optimal among the algorithms with re- stricted number of open bins.
To describe the subclasses of semi-on-line bin packing al- gorithms, we need to introduce the role of the “scheduler” and the “packer”. The role of the scheduler is to produce the input list, while the role of the packer is to pack the items (that is, to implement the packing algorithm).
In the above-mentioned problem classes, the role of the sched- uler is trivial: to supply the elements one by one and to mark the end of the (whole) list.
Gutin et al. introduced [22] the so-called batched bin packing
problem (BBPP) in 2005, which is a semi-on-line bin packing
problem. The classical problem is modified in such a way that the input is split into parts (called batches) by the scheduler. For each step the scheduler either gives a new element, or marks the end of the current batch. Each batch
(that is, a lookahead is possible within the current batch). But lookahead is not allowed outside the current batch, so each batch has to be packed in an on-line manner, where during the packing of a new batch the elements of the earlier packed batches cannot be moved. One batch can consist of more elements or can be empty. If every batch has exactly one element, then we get the classical on-line bin packing problem as a special case. If an input consists of exactly m batches, then we call this BBPP as m-BBPP. The ACR of the algorithms for this problem version can be defined the same way as before. Gutin et al. study the 2-BBPP problem and its variants in [22] in detail. For the 2-BBPP problem, they proved an 1.3871... lower bound here and they derived bounds for those special cases of the 2-BBPP problem, where the number of the different sizes of the elements of the list are bounded from above by a given p ≥ 2 positive integer. They proved, using the results of their paper [23], that if p = 2 then their bound is optimal. They raise an open question of whether for different p(> 2) values their bound is optimal or not.
There is a connection between bin packing and memory al- location scheduling task of computer programs (jobs). The size of a job corresponds to the size of an element, while a memory partition corresponds to the capacity of a bin. The most important difference is that a memory area ordered to a job can be released after finishing the job. To model this, a new class of bin packing algorithms can be defined; the al- gorithms for dynamic bin packing problem (DBP), where the scheduler can specify the arrival of an element (Insert op- eration) or removal of a previously inserted element (Delete operation). Each step of the input is one of these specifica- tions, thus an input list is a finite series of Insert and Delete operations. We should add that a removal can only be a part of the input, i.e. the algorithm, or the packer cannot delete an element; only the input provider (the scheduler) can do this. The number of bins used by a dynamic bin packing algorithm A can be defined as the maximum of the nonempty bins used during the steps. Then the asymptotic competitive ratio R(A) of a dynamic bin packing algorithm
A can be defined in a similar way as earlier by formula (1).
It is not hard to see that the original bin packing problem is the special case of the dynamic problem, where the list consists of inserts only. The dynamic bin packing problem was defined and analyzed by Coffman, Garey and Johnson in [7]. They gave and analyzed approximation algorithms for the problem.
When the scheduler can apply only Insert operations (no Delete) with repacking, Ivkoviˇc and Lloyd proved in [26] that for any ε > 0 there is a (1 + ε)-competitive approxi- mation scheme A, that requires O(log n) amortized time per Insert operation and there is a (1+ε)-competitive fully poly- logarithmic approximation scheme A that requires O(log2n)
amortized time per Insert operation.
For the same bin packing problem, Epstein and Levin [12] gave an APTAS. In their model the total size of the elements moved per step (Insert operation) is bounded by β times the size of the arriving element. Their Algorithm Dynamic
called fully dynamic bin packing (FDBP, [24, 25]) problem. The difference between this and dynamic bin packing prob- lem is that the packer is allowed to perform repacking as well. If the repacking is restricted, i.e. for each step the packer can repack at most c items, then it is called c-repacking fully dy-
namic bin packing problem (c-repacking FDBP). Here c is a
fixed positive integer. Obviously, the case c = 0 is simply the pure on-line bin packing problem.
The classical on-line bin packing can also be relaxed by al- lowing the repacking of at most c elements for each step. This version of the problem is called c-repacking semi-on-line bin packing (c-repacking SOL). Obviously, the case c = 0 again gives the pure on-line bin packing problem.
Many years after Galambos’s first paper on repacking al- gorithms, Gambosi et al. [18, 19] returned to the analy- sis of certain semi-on-line algorithms. Their algorithm used repacking, but the application and the cost of the algorithm was defined in a special way. Their method packs the large elements one by one, while it composes bundles (groups) of the small items. Then one group is moved in one step, and this kind of movement counts as 1-repacking. In this sense these algorithms may move even O(n) items in one step. In paper [19] the authors analyzed two algorithms. The faster, linear time algorithm A1has an ACR of 3/2, while the other
algorithm A2 with O(n log n) running time has an ACR of 4/3.
Ivkoviˇc and Lloyd investigated the FDBP problem. Their algorithm, similar to the technique presented in [19], uses the bundle technique for the small items. Its ACR was 5/4 [25].
Up to the last decade no lower bounds were given for the ef- ficiency of the semi-on-line algorithms. The first paper from this point of view was published by Ivkoviˇc and Lloyd in 1996 [24]. They proved that there is no c-repacking FDBP algorithm which has a better asymptotic competitive ratio than 4/3. With a small modification the construction can be applied for the c-repacking SOL problem as well. This means that the lower bound 4/3 becomes valid for this prob- lem, as was mentioned in [10] by Csirik and Woeginger. Note that the bound of 4/3 is valid for both problems for any c.
2.1 Our earlier results
In [3], we improved the best-known lower bound from 1.3333 [24] to 1.3871 for the c-repacking semi-on-line problem. We presented our proof for the c-repacking SOL problem, and we showed that it remains valid for the c-repacking FDBP problem as well. The results obtained are valid for any c. We proved the lower bound by analyzing and solving a specific optimization problem. For the analysis of the construction we used different methods: LP-techniques were combined with results from linear algebra, and then we solved and analyzed a special non-linear optimization problem. We ex- pressed the exact value of the lower bound in terms of the Lambert W function. Note that our construction is a gen- eralization of the constructions described in [22] and [24].
mal number of the different elements is restricted by a given
p(p ≥ 2) constant. The lower bounds are valid for the spe-
cial cases of both above-mentioned semi-on-line problems. Furthermore, they are valid for the 2-BBPP problem de- fined in [22]. The bounds improve the lower bounds for the case p ≥ 3 given in [22] (and they are valid for both the
c-repacking and classical versions of the problem). Hence
we answered the open question in [22] for the 2-batched bin packing problem, allowing at most p different item sizes. In [2] we proved that our construction can improve these bounds, giving a negative answer for their optimality. The construction works for specific p (p > 2), and the lower bounds are valid for the above two problems, the c-repacking SOL problem and the c-repacking FDBP.
In [4] we proved that although our bounds are given in one dimension, the above-mentioned lower bounds are valid for every d-dimension (d ≥ 1) for multidimensional (geomet- ric) semi-on-line bin packing problems. To the best of our knowledge this is the first multidimensional semi-on-line bin packing result. The lower bound construction is given by hypercubes and so it is valid for the classical d-dimensional on-line hypercube packing problem, even when repacking is not allowed. The bound improves the lower bound of 4/3 ([8]), which was the best known lower bound for cases d ≥ 4. The interesting aspect of the lower bounds presented in [2] and [3] is that although the problem is clearly discrete, the solution requires that one solve specific nonlinear (continu- ous) optimization problems. To do this, we applied global optimization methods. So the construction furnished a nice connection between combinatorial and global optimization. In [2], we solved the nonlinear optimization problem by a reliable Branch and Bound method based on interval arith- metic [30, 31]. Reliability means that the proof is produced by a computer, but the results are checked and the inaccu- racy arising from rounding errors is eliminated.
In another paper [5], we improved our previous results and gave new upper bounds for the c-repacking SOL problem. We analyzed our algorithm using classical methods (like the weighting function technique). We gave a series of algorithms HFR-c for any positive integer value c. We proved that if c goes to infinity, then R(HFR-c) goes to 3/2. More precisely, we proved that the ACR for a given c is not larger than 3/2 + bc/(1 − bc), where bc is in the interval (0, 1/(6c)]. The given upper bounds show that repacking really can help. To demonstrate this we state some cases of these. In the case of c = 1, the result of our algorithm is irrelevant from the point of view that the best on-line algorithm is more competitive [34]. However, our 1-repacking algorithm uses far fewer bin classes. The ACR of our algorithm HFR-2 for the case of c = 2 is smaller or equal to 1.5728..., which is less than the best known ACR for on-line algorithms [34]. Another interesting aspect of the case c = 2 is that the given ACR is smaller than the lower bound 1.58333 proved for Harmonic Fit type on-line algorithms [32]. The same holds for our algorithm HFR-3 in the case c = 3: R(HFR- 3)≤ 1.5507.... Another important issue is that for the case of c = 4 our algorithm has a better ACR than the best lower bound for on-line algorithms [1]. This means that it is more
This shows us that it is worth investigating such algorithms.
Many open issues remain, of which we shall mention three. The first is to give a 1-repacking algorithm with an ACR better than 1.58333. The second is to provide a semi-on-line
c-repacking algorithm with an ACR better than 1.5403 —
for c < 4. The third is to improve the lower bounds for small
c values (c = 1, 2).