A tight lower bound for the online bounded space hypercube bin packing problem

In the d-dimensional hypercube bin packing problem, a given list of d-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio ρ of the online bounded space variant is Ω(log d) and O(d/log d), and conjectured that it is Θ(log d). We show that ρ is in fact Θ(d/log d), using probabilistic arguments.

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1 Introduction

The bin packing problem is an iconic problem in combinatorial optimization, which has been investigated intensively from many different viewpoints. In particular, it has served as a proving ground for new approaches to the development and analysis of approximation and online algorithms, as well as for the development of average case analysis techniques (see

Coffman Jr. et al. (1997, 2013)).

We prove a lower bound for a variant of the bin packing problem, in which the items to be packed are -dimensional hypercubes, also referred to as -hypercubes or simply hypercubes, when the dimension is clear. More precisely, we prove a tight lower bound for the online bounded space -hypercube bin packing problem, settling an open problem raised by Epstein and van Stee (2005). Before we state our result (Theorem 5), we introduce the required concepts and definitions and discuss briefly the relevant literature.

The -hypercube bin packing problem (-CPP) is defined as follows. We are given a list of items, where each item  in  is a -hypercube of side length , and an unlimited number of bins, each of which is a unit -hypercube (that is, a -hypercube of side length ). The goal is to find a packing  of the items in  into the smallest possible number of bins. More precisely, we have to assign each item to a bin, and specify its position in that bin. We require that the items be placed parallel to the axes of the bin and, crucially, we require that the items in a bin should not overlap. The size  of the packing  is the number of used bins (those with assigned items).

The -hypercube bin packing problem (-CPP) is in fact a special case of the -dimensional bin packing problem (-BPP), in which one has to pack -dimensional parallelepipeds into -dimensional unit bins. For , both problems reduce to the well known bin packing problem.

In the online variant of -CPP, the items arrive sequentially and each item must be placed in some bin as soon as it arrives, without knowledge of the next items. The online bounded space variant of -CPP is a restricted variant of online -CPP. Whenever a new empty bin is used in the packing process, it is considered an open bin and it remains so until it is considered closed, after which point it is not allowed to accept other items. In this variant, regardless of the instance , at every point of the process, not more than  bins should be open, where  is some constant that does not depend on .

As usual for bin packing problems, we consider the asymptotic performance ratio to measure the quality of algorithms. For an algorithm  and an input list , let  be the number of bins used by the solution produced by  for the list . Furthermore, let , where the minimum is taken over all possible packings  of  into unit bins. The asymptotic performance ratio of  is

(1)

Given a packing problem , the optimal asymptotic performance ratio for is

(2)

Many results have been obtained for online -BPP and -CPP (see, e.g., Balogh et al. (2019, 2012); Blitz et al. (2017); Christensen et al. (2017); Heydrich and van Stee (2016); Seiden (2002); van Vliet (1992)). In our brief discussion of the literature below, we restrict ourselves to the online bounded space versions of -BPP and -CPP.

For online bounded space -BPP, Lee and Lee (1985) gave an algorithm called  with asymptotic performance ratio at most , where  as , and  is a certain explicitly defined constant. These authors also proved that no algorithm for online bounded space -BPP can have asymptotic performance ratio smaller than . For online bounded space -BPP for general , a lower bound of  was implicitly proved by Csirik and van Vliet (1993), and Epstein and van Stee (2005) proved an asymptotically matching upper bound.

For online bounded space -CPP, Epstein and van Stee (2005) proved that its asymptotic performance ratio is  and , and conjectured that it is . They also gave an optimal algorithm for this problem, but left as an interesting open problem to determine its asymptotic performance ratio. Later, Epstein and van Stee (2007) gave lower and upper bounds for .

Our main contribution is an lower bound for online bounded space -CPP. In view of previous results by Epstein and van Stee (2005), we obtain that the asymptotic performance ratio of this problem is , settling an open problem posed by those authors. To prove our lower bound, we follow a well known approach (see  Lee and Lee (1985) and Yao (1980)), which requires the proof of the existence of a packing with a suitably large ‘weight’, for a certain definition of weight. The novelty here is that we prove the existence of such a packing with the probabilistic method.

To conclude this section, we mention that the technique that we present here may also be used to obtain lower bounds for the prices of anarchy of a game theoretic version of -CPP, called selfish -hypercube bin packing game. As this topic requires the introduction of a number of concepts, we just mention the main results for readers familiar with this line of research: for every large enough , the asymptotic price of anarchy (respectively, strong price of anarchy) of the selfish -hypercube bin packing game is (respectively, ). The proof of one of the results can be found in Kohayakawa et al. (2017). A preliminary version of this work (Kohayakawa et al. (2018)) appeared in the proceedings of LATIN 2018.

2 Notation and homogeneous packings

The -hypercubes  defined below will be important in what follows.

Definition 1.

Let  be an integer. For every integer  and , let

(3)

be the open -hypercube of side length  ‘based’ at the origin.

2.1 Homogeneous packings

We shall be interested in certain types of packings of hypercubes into a unit bin.

Definition 2 (Homogeneous packings ).

Let  be fixed. For any integer and , a packing of copies of  into a unit bin is said to be a packing of type . Packings of type  will be called homogeneous packings.

In the definition above, the upper bound on  guarantees that  copies of  can be packed into a unit bin (and hence  exists): it suffices to note that, under that assumption on , we have . Homogeneous packings are important because they can be used to create instances for which any bounded space algorithm performs badly (see Epstein and van Stee (2005, 2007)).

3 The central lemma and the main theorem

The key result used in the proof of our main theorem (Theorem 5) is the existence of a certain packing, stated in Lemma 4 below. Since this lemma is somewhat technical, we first informally describe a related result.

Consider the  homogeneous packings  (), where  for a small positive constant . Suppose also that  for some small . Suppose we assemble a list  of -hypercubes from these  homogeneous packings  by selecting  of the members of each such . The following holds: (*) there is a packing of  into a single unit bin as long as  is sufficiently large. This fact is behind the proof of our central lemma, Lemma 4, stated in what follows. Fact (*) might look surprising at first sight, as the homogeneous packings  appear to have reasonably high occupancy.

We now give some definitions needed for the statement of Lemma 4.

Definition 3 (-packings).

A packing  of -hypercubes into a unit bin is called an -packing if, for every member  of , there is some integer  such that  is a copy of .

Let  be an -packing for some . Let

(4)

For every , let

(5)

Clearly, we have  for every  (recall that ). Finally, we define the weight of  as

(6)

We shall be interested in -packings  with large weight. Our main lemma is as follows.

Lemma 4 (Central lemma).

There is an absolute constant  for which the following holds for any . For any , the unit bin admits an -packing  with

(7)

In (7) and in what follows, stands for the natural logarithm of . The proof of Lemma 4 is postponed to Section 4. We now deduce our main result, Theorem 5, from Lemma 4, following the approach used by Lee and Lee (1985). For experts in the area, given Lemma 4, the proof of Theorem 5 is routine. The short proof below is included for the benefit of non-experts.

Theorem 5 (Main Theorem).

There is an absolute constant  such that, for any , the asymptotic performance ratio of the online bounded space -hypercube bin packing problem is at least .

Proof.

Let  be any algorithm for the online bounded space -hypercube bin packing problem. Let  be the maximum number of bins that  leaves open during its execution. To prove that  has asymptotic performance ratio at least  if  is large enough, we construct a suitable instance  for .

Let  be as in Lemma 4 and suppose . Fix any  with  and let  be a packing as in the statement of Lemma 4. The instance  will be constructed as follows. First, we choose a suitable integer  and take  copies of . We then construct  by arranging the hypercubes in these copies in a linear order, with all the hypercubes of the same size appearing together. Let us now formally describe .

Let . Recall that  contains  copies of  for every . Let  and suppose . The instance  that we shall construct is the concatenation of  segments, say , with each segment  () composed of a sequence of  copies of . This completes the definition of our instance .

The following assertion, to be used later, concerning the offline packing of the hypercubes in  is clear, as we obtained  by rearranging the hypercubes in  copies of .

The hypercubes in  can be packed into at most  unit bins. (8)

We now prove that, when  is given the instance  above, it will have performance ratio at least as bad as . In view of (7) in Lemma 4, this will complete the proof of Theorem 5.

Let us examine the behaviour of  when it is given input . Fix  and suppose that  has already seen the hypercubes in  and it has already packed them somehow. We now consider what happens when  examines the  hypercubes in , which are all copies of .

Clearly, since , the  copies of  in  cannot be packed into fewer than

(9)

unit bins. Therefore, even if some hypercubes in  are placed in bins still left open after the processing of , the hypercubes in  will add at least  new bins to the current output of . Thus, the total number of bins that  will use when processing  is at least

(10)

In view of (8), it follows that the asymptotic performance ratio of  is at least

(11)

as claimed. This completes the proof of Theorem 5. ∎

4 Proof of Lemma 4

The -packing  whose existence is asserted in our central lemma, Lemma 4, will be described in terms of certain ‘codes’, that is, sets of ‘codewords’ or simply ‘words’. We shall use such codes to ‘place’ copies of certain  in the packing . We make this precise in Section 4.1. The proof of the existence of appropriate codes will be given in Section 4.2. The proof of Lemma 4 is given in Section 4.3.

4.1 Placing hypercubes according to codewords

Let  and  be fixed. Let a -letter word  from the alphabet  be given. In what follows, we shall fix some  and we shall consider translations  of the hypercube  specified by such words  in a certain way (for the definition of , recall (3)). Furthermore, later, we shall consider certain sets  of such words and we shall define packings of the form . Note that  is composed of copies of . To obtain the packing  whose existence is asserted in Lemma 4, we shall consider the union of such packings  for , with  and certain families  (see Lemma 13).

Let us now define , the translation of  specified by . For  with  for every , we let  be the translation

(12)

of , where

(13)

Thus, while  has its ‘base point’ at the origin,  has its base point at  (see  and  in Figure 1).

In what follows, we shall always have . Therefore, if  for every , then  is contained in the unit hypercube , whereas if  for some , then with  as defined in (13) is not contained in  (see  in Figure 1). Since we want  to be contained in  for every , we actually define  as in (15) below.

Definition 6 (Base point coordinates of ).

For every  and , let

(14)

For , let

(15)

Finally, for convenience, for , let

(16)

Figure 1: Projections on the -plane of hypercubes , and  with , and , and  and . The hypercube  is not contained in .

We now state three simple facts that the reader may find useful to check on their own to get used to the definitions above. First, note that is a packing of  copies of  into the unit bin ; that is,  is a packing of type  (recall Definition 2). Secondly, is not a packing. Finally, is a packing (and is also a packing of type ).

Note that, because , for every , we have

(17)

(see Figure 1). For every  and , let

(18)

Finally, note that

(19)

We close this section observing the following.

Fact 7.

The following assertions hold for any positive .

  1. Suppose  and . Then

    (20)

    In particular, the intervals  are disjoint from .

  2. For any , the intervals  are pairwise disjoint, except for the single pair formed by  and .

Proof.

Assertion (ii) is clear (recall (17)). The second assertion in (i) follows from inequality (20), and therefore it suffices to verify that inequality. We have . Moreover, . Therefore, (20) is equivalent to

(21)

Since  and , inequality (21) does hold. ∎

4.2 Separated families of gapped codes

Let an integer  be fixed. We shall consider sets of words  for . We refer to such  as codes or -codes. As discussed in the beginning of Section 4.1, we shall design such  to specify packings .

We start with the following definition.

Definition 8 (Gapped codes).

Suppose  and let a -code be given. We say that  misses  at coordinate  if every word  in  is such that . Furthermore,  is said to be gapped if, for each , either  misses  at  or  misses  at .

Suppose  is a gapped code, and suppose  and  are distinct words in . Then  and  do not overlap: this can be checked from (19) and Fact 7(ii). Thus, if  is gapped, then

(22)

is a packing.

We now introduce a certain notion of ‘compatibility’ between two codes  and , so that  and  can be put together to obtain a packing if they come from ‘compatible’ codes  and .

Definition 9 (Separated codes).

Suppose  and  and  are given. We say that  and  are separated if, for any  and any , there is some  such that .

Suppose  and  are gapped and separated and suppose  and  for some  (we shall later set  to be a certain value ). Consider the packings  and  as defined in (22). Fact 7(i) and (19) imply that  is a packing. Indeed, let  and any  be given. Then, by definition, there is some  such that . This implies that  and  are disjoint ‘in the th dimension’ (see Fact 7(i)).

Definition 10 (Separated families).

Let  be a family of -codes . If, for every , the codes  and  are separated, then we say that  is a separated family of codes.

Remark 11.

For , let . Then  is a separated family of gapped codes. Fix . Consider  with  as in (22). Since each  is gapped, the  are packings. Also, since  is a separated family,  is a packing. Furthermore, we have  (recall (5)) and  (recall (6)). The existence of  implies a weak form of Theorem 5 (namely, a lower bound of  instead of ).

Remark 11 above illustrates the use we wish to make of separated families of gapped codes. Our focus will now shift onto producing much ‘better’ families than the one explicitly defined in Remark 11. Indeed, we now prove Lemma 13 below, which asserts the existence of such better families. We shall need the following auxiliary lemma.

Lemma 12.

There is an absolute constant  such that, for any , there are sets  such that (i) for every , we have  and (ii) for every , we have .

Proof.

Let . We select each  () among the -element subsets of  uniformly at random, with each choice independent of all others. Let . Note that, for any , we have . Let . Let

(23)

as long as 

is large enough. We may now apply a Chernoff bound for the hypergeometric distribution (see,

e.g., Janson et al. (2000), Theorem 2.10, inequality (2.12)) to obtain that

(24)

for every large enough . Therefore, the expected number of pairs  with for which  is less than , which tends to  as . Therefore, for any large enough , a family of sets  as required does exist. ∎

We are now ready to state and prove the lemma that asserts the existence of a separated family of gapped codes that is ‘better’ than the one defined in Remark 11.

Lemma 13 (Many large, separated gapped codes).

There is an absolute constant  such that, for any , there is a separated family  of gapped -codes  such that

(25)

for every , where

(26)
Proof.

Let  be as in (26) and let  be as in Fact 12. In what follows, we only use the  for . For each , we construct  in two parts. Suppose first that we have  with

(27)

We then set

(28)

Note that, by (27) and (28), the -code  will be gapped ( is missed at every  and  is missed at every ). We shall prove that there is a suitable choice for the  with , ensuring that  is separated. Since we shall then have

(29)

condition (25) will be satisfied and Lemma 13 will be proved. We now proceed with the construction of the codes  ().

Fix . For , let , and note that

(30)

Let  be an element of  chosen uniformly at random. For every , let us say that  is -bad if  for every . We have

(31)

for every large enough . Let us say that  is bad if it is -bad for some . It follows from (31) that

(32)

if  is large enough. Therefore, at least  words  are not bad, as long as  is large enough. We let  be the set of such good words.

To complete the proof, it remains to show that the family is separated. More precisely, we show that with the above choice of  , the family  with  as defined in (28) is separated.

To this end, fix . We show that  and  are separated. Let  and  be given. By the definition of , there is  such that  for all . Furthermore, since  is not a bad word, it is not -bad. Therefore, there is  for which we have . Observing that  and recalling the definition of , we see that , as required.

The proof of Lemma 13 is now complete. ∎

4.3 The packing in Lemma 4

Fix , a separated family of gapped -codes . We now give, for every sufficiently small , the construction of a packing  of -hypercubes into the unit bin  using  and prove that  is indeed a packing. Choosing  as in Lemma 13 above, we shall deduce Lemma 4 by taking .

Definition 14 (Packing ).

Suppose is a separated family of gapped -codes . Let . We put

(33)

where  is as in (22).

In Lemma 15 below, we compile the properties that we need of . For the relevant notation, recall (4), (5) and Definition 3.

Lemma 15.

Suppose is a separated family of non-empty gapped -codes . Suppose . Let   be the family of all the hypercubes  with  and . Then the following assertions hold: (i) the hypercubes in  are pairwise disjoint and form an -packing; (ii) for every , we have ; (iii) .

Proof.

Let us first check that the hypercubes  in  are pairwise disjoint. We remark that, when introducing the notions of gapped and separated codes, we already discussed the reason why the  in  are indeed pairwise disjoint. However, we give a formal proof here for completeness. Let  and  with  be given. Consider  and . We have to show that

(34)

Suppose first that . In that case, both  and  are in  and we may suppose that . Thus, there is some  such that . Furthermore, since  is gapped, either  or  is missed by  at . In particular, the pair  cannot be the pair  and therefore

(35)

(recall Fact 7(ii)). Expression (19) applied to  and , together with (35), confirms (34) when .

Suppose now that . Since  and  are separated, there is some  such that . Fact 7(i) tells us that

(36)

Expression (19) applied to  and , together with (36), confirms (34) in this case also. We therefore conclude that  is indeed a packing.

The hypercubes in are copies of the hypercubes  for , and therefore is an -packing. This concludes the proof of Lemma 15(i). Assertions (ii) and (iii) are clear. ∎

We are now ready to prove Lemma 4.

of Lemma 4.

Let  be as in Lemma 13. We may and shall suppose that  and that  is large enough so that, for every , the last inequality in (37) below holds. We prove that Lemma 4 holds with this choice of . Let  and  be given. Let . Note that