Tight Bounds for Online Vector Bin Packing

Transcription

1 Tight Bouns for Online Vector Bin Packing Yossi Azar Tel-Aviv University Seny Kamara Microsoft Research, Remon, WA Ilan Cohen Tel-Aviv University Bruce Shepher McGill University, Montreal, Canaa Abstract In the -imensional bin packing problem (VBP), one is given vectors x 1,x 2,...,x n R an the goal is to fin a partition into a minimum number of feasible sets: {1,2...,n} = s i B i. A set B i is feasible if j Bi x j 1, where 1 enotes the all 1 s vector. For online VBP, it has been outstaning for almost 20 years to clarify the gap between the best lower boun Ω(1) on the competitive ratio versus the best upper boun of O(). We settle this by escribing a Ω( 1 ε ) lower boun. We also give strong lower bouns (of Ω( 1 B ε ) ) if the bin size B Z + is allowe to grow. Finally, we iscuss almost-matching upper boun results for general values of B; we show an upper boun whose exponent is aitively shifte by 1" from the lower boun exponent. 1 Introuction We stuy the vector bin packing problem (VBP) which has receive renewe attention in connection with research on virtual machine placement in clou computing, e.g., [23, 18, 22]. In the offline version, one is given a collection V = {x 1,x 2,...,x n } of vectors x i [0,1], an a subset X V is a feasible bin if xi X x i 1. If the bins have size B, then feasibility asks for xi X x i B (bol face inicates a -imensional vector). We enote by OPT (V ) the minimum number of bins neee to partition the vectors into feasible bins. If all vectors are binary, we refer to this as {0,1} VBP. We also sometimes refer to the packing integer program (PIP) version of the problem, where we seek the largest set of vectors which fits into a single bin. In the online version, the vectors arrive one by one, an at the time of arrival, the algorithm A must either assign a vector to an open bin, or open a new bin to which it is assigne (maintaining bin feasibility of A(V ) course). The algorithm s competitive ratio on an instance V is OPT (V ), where A(V ) enotes the number of bins opene by A. Similarly, its competitive ratio on a class of instances is the worst ratio on an instance in the class; its overall competitive ratio is this measure taken on the class of all -imensional instances of VBP. Supporte in part by the Israel Science Founation (grant No. 1404/10). Part of this work was one while the author was visiting Microsoft Research, Remon. Supporte in part by the Google Inter-university center. Supporte in part by Natural Sciences Engineering Research Council of Canaa 1

2 While there is a simple first-fit O()-competitive algorithm for VBP (see [11]), the best lower bouns have been constant. Specifically, the bouns from [9] are at most 2. As pointe out in [10], this gap has persiste, an in fact in [7] it is conjecture to be super-constant, though sublinear. We show (Section 2) a lower boun of Ω( 1 ε ) for any ε > 0 giving an essentially tight result. The arguments make critical use of techniques from [14] for online colouring an [5] for stochastic PIPs. We also nee further structural insights into the class of graphs use by [14] - see Section 2.1. Our main lower boun results are the following. Theorem 1.1. For any integer B 1, any eterministic online algorithm for VBP has a competitive ratio of Ω( 1 B ε ). For {0,1} VBP the lower boun is Ω( 1 B+1 ε ). These lower bouns are information-theoretic an hence apply also to exponential time algorithms. These bouns are tight for B = 1 since there is a ( +.7)-competitive algorithm for V BP [11]. In fact for B = 1, this settles a problem pointe out in [10]: the main open problem in on-line -imensional vector packing consists in narrowing the wie gap between 2 an + 7/10. However, for B 2, there was a gap in our unerstaning. For instance, for {0,1} VBP an B 2, it is known that the greey algorithm is 2B -competitive. This upper boun exponent of 1 2 is off from our lower boun exponent of 1 B+1 (which seems the right answer). We partially rectify this in Section 3 via the following upper boun. Theorem 1.2. There are online V BP algorithms for B 2 with competitive ratio: O( 1/(B 1) log B/(B 1) ), for [0,1] vectors. O( 1/B log (B+1)/B ), for {0,1} vectors. Note that for B log the boun becomes O(log). It is worthwhile to mention that there is still an aitive shift by 1" gap in the exponent. Specifically, for {0,1} vectors, the exponent of the upper boun is 1/B whereas the exponent of the lower boun is 1/(B + 1). For general vectors the ifference in exponent is 1/(B 1) versus 1/B. 1.1 Techniques For the lower boun, a core iea that we use is a lower boun strategy for online colouring that has been aroun for some time [14]. They show that there are graphs with chromatic number O(log n) for which any online algorithm may use up to n/logn colours. In the online colouring moel, an aversary prouces a sequence of noes v 1,v 2,...,v n such that upon arrival, v i reveals its neighbours amongst v 1,v 2,...,v i 1. Moreover, in their moel the aversary is transparent in that it reveals the actual colour of v i just after the online algorithm makes its choice. Strangely, this extra feature is important for our arguments to work. Similar bouns for colouring are given in [13] for a stronger moel, where the graph being coloure is reveale to the algorithm at the beginning (but the aversary selects a subgraph to be coloure). We refer the reaer to the surveys of online packing (an covering) [10] an [4] for further backgroun. For the upper boun we combine some state of the art techniques. The algorithm we present is base on two parts. The first part solves V BP using a loa balancing technique with a set of virtual enlarge bins (of size cblog). The algorithm ecies on the assignments using exponential weights for each virtual bin. The secon part assigns vectors of each virtual bin to real bins of size B. For the secon part we present two algorithms. First, a simple ranomize algorithm, an we use Chernoff bouns to compute the expecte number of bins. Secon, a eterministic algorithm which is a e-ranomization of the ranomize algorithm using techniques from [1]. Combining both parts prouces a V BP algorithm with the esire performance. 2

3 1.2 Relate Work VBP has classical connections to scheuling problems where jobs nee to be run on machines, each with a boune supply of some resources (e.g., CPU time, memory etc). Such applications have been actively stuie recently ue to interest in clou computing, e.g., the problem of virtual machine placement an migration, cf. [18]. Epstein [7], largely motivate by the gap for online VBP, initiate the stuy of -imensional vector packing with variable size bins. In this case, one is given a collection B of bin capacity profiles (each a vector in R ), where the collection is assume to contain 1. In that setting, they can show that for certain choices of B, one may obtain linear lower bouns, whereas for other choices, the competitive ratio can be at most 1 + ε (the choice of B epens on ε). They conjecture that in the classical case, where B = {1}, the competitive ratio is super-constant, but sublinear. When = 1, variable-size bin packing was introuce by Friesen et al. [8] an subsequently stuie in [19]. There is also extensive work on (polytime) approximation algorithms for offline VBP cf. [3]. We mention that until recently there was a gap between Ω(.5 ε ) an O( 1 ε ) for the approximation version of offline PIP. This was settle in [5] in their stuy of stochastic PIPs. In [2] it was shown that for any ε > 0, a polytime.5 ε -approximation for VBP woul imply NP=ZPP. By using techniques of [6] it is not har to strengthen this lower boun to 1 ε ; the arguments become apparent in Section 2.2. Proposition 1.3. For any ε > 0, a polytime 1 ε -approximation for VBP implies that NP=ZPP. In contrast, our lower boun is information-theoretic an hols without any complexity assumptions. In particular it applies also to exponential time algorithms. We close by remarking that while online packing (PIPs) are structurally very relate to online VBP, lower bouns for the packing version are more immeiate. For instance, the online maximum stable set problem (in the setting escribe above for online colouring) is completely trivial. This is in stark contrast to the relatively rich theory for online colouring (cf. [20, 12, 15, 17]). A trivial lower boun of n is achieve as follows. As soon as the online algorithm puts some noe in its stable set, the aversary makes all future noes ajacent to the selecte noe. This argument is not quite correct for online PIPs, since the aversary must reveal the whole column being packe (i.e., ajacencies to future noes as well). See Appenix A for a slightly more elaborate an vali argument. 2 Lower Bouns A core iea in the first step of our lower boun proof employs arguments from [14] which shows a lower boun for online colouring. Importantly for us, their results hol for the following class of graphs, which we call the HS graphs. An HS graph G of orer k (assume k even) is a graph whose noes are a sequence (v 1,v 2,...,v n ) with the following aitional properties. G has a k-colouring c : V [k] which is compatible with some amissible colour assignment. Each noe v has been assigne a k/2 subset F(v) of [k] (its inamissible set) an for each u v (in the sequence), uv E(G) if an only if c(u) F(v). We call this extra property consistency. The number of noes in such an HS graph is assume to be n = k 2 (k k2 ), an so k = O(logn). In particular, it is important to note that consistency implies: if v is ajacent to some noe of colour c before it in the sequence, then it is ajacent to all noes of colour c before it. In their work, the aversary prouces a sequence of n noes v 1,v 2,...,v n of some HS graph. When it announces v i, it must share all ajacencies to earlier noes in the sequence. The aversary can even 3

4 n 1 B+1 ɛ (B + 1)-free graphs Aversary s Colouring G 1 G k + 1 B-free subgraphs of G (B 1)-free Graphs Last Layer noe s final colour G announce k an n ahea of time. Moreover, it taunts the online algorithm in the following sense: after the online algorithm colours v i, the aversary announces its real colour c(v i ). These aitional properties are referre to as a transparent aversary. They establish that there are such graphs with k = O(logn) for which any online algorithm will use Ω( n logn ) colours. That is, the aversary may force the algorithm to use this many colours. Their elegant result is succinctly capture in the following. Theorem 2.1 ([14]). For any online colouring algorithm, its competitive ratio on the class HS is at least 2n/log 2 n, even against a transparent aversary. Specifically, any online colouring algorithm uses at least 2n/logn colours, where any graph in the HS class is logn coloure. 2.1 The {0,1} Case We now establish the secon half of Theorem 1.1. This boun is tight in the B = 1 case, an almost-tight for B > 1. (See Section 3.) Theorem 2.2. For any fixe ε (0,1), any online {0,1} VBP algorithm has competitive ratio at least 1 B+1 ε. Proof. Theorem 2.1 strongly suggests a reuction strategy to obtain lower bouns for {0, 1} VBP. We follow this approach, an show how to convert an online stream of noes in an HS graph, into an online stream of vectors for VBP. We o this so that if there is a VBP algorithm with competitive ratio O( B+1 1 ε ), then there is an online colouring algorithm with competitive ratio O(n 1 ε ), thus contraicting Theorem 2.1. There are two complications. The first is to eal with the fact that in VBP, the online algorithm has complete information about incoming vectors, whereas in online colouring, we only know about ajacencies to noes which arrive earlier. We can eal with this by expaning the imension of the space where packing occurs from = n to = n B+1. Partly as a warm-up an partly for peagogy, we escribe this for the case B = 1. The ieas are core to the more complicate arguments for larger B. 4

5 Consier a stream of noes arriving from an HS graph. For each new noe v i, we supply the online VBP algorithm a 0,1 vector x i with n 1 s, an (n 2 n) 0 s. These form the columns of a matrix A that is graually built by the algorithm. For each j < i, if v i v j is an ege of the HS graph, then fin a row which has exactly one 1 in column j. The vector x i (i.e., next column of A) will also inclue a 1 in this row. If v i saw r earlier noes, then we place r 1 s in this fashion. We then place (n r) 1 s in fresh rows, i.e., rows such that A oes not yet contain any 1 s. These will be use for later noes which are ajacent to v i. Let A be the final matrix constructe, an note that if Az 1 for some z {0,1} n, then z must be the incience vector of a stable set. Hence the optimal offline solution for VBP is precisely χ(g), the chromatic number of G. Since the number of rows = n 2, we have the following. If there is an online algorithm for {0,1} VBP with competitive ratio at most.5 ε, then it woul prouce a colouring of G that uses at most n 1 2ε colours. For any fixe ε > 0, this contraicts Theorem 2.1. Note that this actually completes the proof in the case when B = 1. For B 2, there are further complications. First, we use a construction from [6]. This helps us convert a stream of noes in an HS graph, into a stream of vectors with the following property. A vali online VBP will pack these vectors into bins such that each bin correspons to a K B+1 -free graph of the HS graph. The secon issue is to turn this into a goo colouring of the original graph. To each K B+1 -free graph, we apply a secon online filtering algorithm to prouce a goo colouring. Interestingly, we make critical use of both the structure of HS graphs, as well as the fact that the bouns of [14] are with respect to a transparent aversary. We nee to know the earlier colours! Again, for simplicity we start by explaining the argument for the case B = 2. Suppose that we have an online algorithm for {0,1} VBP with B = 2 which is B+1 1 ε -competitive for ε (0,1). Consier the noes of an HS graph G arriving v 1,v 2,...,v n. To each of these we associate a 0,1 vector x i in R n3 with n 2 1 s an (n 3 n 2 ) 0 s. As before the vector x i must prepare for future ajacencies. Now, it actually prepares for future triangles that contain it. For each triangle v i,v j,v p involving noe i, the algorithm oes the following. If i < j, p (it is the first noe of the triangle to arrive) then it places a 1 in some row of A which oes not yet contain any 1 s; this is possible since there are = n 3 rows. Otherwise, if say j < i, then there is some row which is alreay eicate to this triangle an has a 1 in column j (an column p if p < i). We also place a 1 in this row of vector x i. Consier the final matrix A constructe, an a 0,1 vector z (i.e., selection of columns) such that Az B. Then, following [6], these columns ientify a triangle-free subgraph of G when B = 2 (an a K B+1 -free subgraph if we were preparing vectors for future (B + 1)-cliques more generally). Suppose that there is a B+1 1 ε -competitive algorithm for {0,1} VBP when B = 2. We claim that we can convert this into an n 1 3ε -competitive algorithm for colouring HS graphs, contraicting Theorem 2.1. To see this, we fee the vectors (efine above) to our online VBP algorithm that partitions them online into colour classes G i : i = 1,2...,s which are necessarily triangle-free. Note first that for an HS graph of orer k, the minimum partition into triangle-free graphs is clearly at most k (since it is actually k-colourable). Hence, by the competitive ratio, we have s 3 1 ε k = n 1 3ε k. We now show an online algorithm which converts these triangle-free graphs into real colourings. One easily proves that any triangle-free graph on N noes, has chromatic number at most N. However, we o not know of an online algorithm which guarantees to fin a colouring of size at most N 1 β for any β > 0. Instea, we appeal to the special structure of HS graphs. For each i, we fee the noes of G i to a secon online colouring algorithm. The secon online algorithm sees the noes of G i arrive as w 1,w 2,...,w N, i.e., these are the noes corresponing to the vectors packe into triangle-free class i. The algorithm crucially epens on the fact that when it processes some w j it actually knows the aversary s colouring on noes before it (but not w j itself). If some new noe w j is not ajacent to any earlier noe, we assign it colour 0. Otherwise, let w j 5

6 be the first noe in G i s sequence to which it is ajacent. Assuming a transparent aversary, we also know a colour c = c(w j ) {1,2,...,k} for this noe in G. We then assign the colour c to w j for our own colouring (we will actually assign w j a ifferent colour, possibly 0 for instance). We now argue that this prouces a vali colouring of G i. Note that each noe is either given colour 0 (an these noes form a stable set), or it is sinke into the first noe in the sequence (w 1,...,w N ) to which it is ajacent. Suppose some w i is sinke into the noe w j which has colour c. In fact w j was the first noe in this sequence to get colour c. That is because in HS graphs, a noe is ajacent to some noe of colour c before it in the sequence, if an only if it is ajacent to every noe of this colour before it in the sequence. Hence the noes we coloure c form a star with center w j. Since G i is triangle-free, these noes form a stable set an so we have a vali colouring. Note that we use at most k + 1 colours to online colour G i. In total, we have thus use (k + 1)s (k + 1)n 1 3ε k. We may choose the orer k of our HS graph large enough so that (k + 1)n 1 3ε k < 2n/logn contraicting Theorem 2.1. (Recall k = O(logn) in HS graphs.) We now consier the lower boun for B 3. Here we apply a recursive argument. Consier {0,1} VBP where we now prouce a matrix with = n B+1 rows, one for each (B + 1)-clique. Then a B+1 1 ε - competitive algorithm will partition G into at most n 1 (B+1)ε k graphs, each of which is K B+1 -free. On each such subgraph G i, we run the same online filtering algorithm use above on K 3 -free graphs. As before, each colour class consists of a star, i.e., a set of noes with a common neighbhour in the subgraph G i. This may no longer form a stable set, but it is guarantee to be K B -free since G i is K B+1 -free. We then apply this online algorithm again, on each of the resulting K B -free subgraphs. We call this filtering. Note that a noe w i again only looks at noes w j which arrive before it, but in aition it only consiers noes that ha the same colour in all of the preceing filtering phases. Figure 2.1 gives an example of the filtering process on the first K B+1 -free graph G 1. After filtering B 2 times, we are guarantee to en up with K 3 -free classes. One further roun of filtering then guarantees that the final colour classes are inee stable sets, just as in the B = 2 case. This prouces a colouring of G as follows. If a noe was in G i, an the colours it receive in the B 1 filters (one for each K t -free subgraphs, t = B + 1,B,B 1,...,3) were c B+1,c B,..., then it is assigne colour G 1 c B+1 c B... Since at most k + 1 colours are use at each layer, the total number of colours use is at most (k + 1) B 1 n 1 (B+1)ε. Since B is a constant, we may choose the orer k of our HS graph large enough so that (k +1) B 1 n 1 (B+1)ε < 2n/logn contraicting Theorem 2.1. (Recall k = O(logn) in HS graphs.) 2.2 The [0,1] Case The lower boun for the general case nees some moification. In particular, we employ an iea from [6] that was use in closing the approximation gap associate to the offline version of PIP. We show that a 1 ε -competitive algorithm for VBP woul imply an n 1 ε competitive algorithm for colouring the family of HS graphs. We follow the proof of Theorem 2.2, feeing vectors x i to an online VBP algorithm as we receive noes from some graph G. The vectors now live in [0,1] n. We set the i th component of x i to 1, an for each j < i with v i v j E(G), we place 1 n in the jth component. All other entries are 0. For the final matrix A, we have that Az 1 for a 0,1 vector z, implies that z is the incience vector of a stable set. Hence a vali (online) vector packing again ientifies a vali (online) colouring of G. Since = n, a 1 ε -competitive VBP algorithm, woul prouce a n 1 ε -competitive algorithm for colouring HS graphs, contraicting Theorem 2.1. This establishes a complete proof of the first half of Theorem 1.1 for the case B = 1. In the B = 2 case, we now prouce vectors prepare for future triangles as in the proof of the {0,1} VBP, B = 2 case. However, we are able to o this using only = n 2 imensions (not 3 ) as follows. This combines several ieas we have seen so far. Each vector will have entries in 0,1, 1 n an each row will be 6

7 associate to (at most) one ege. Suppose that v i,v j,v p forms a triangle in G where j < p < i. When j arrives, the vector x j is prepare for the ege jp by placing a 1 in a fresh row. When p arrives, it places a 1 in the same row. In the future, for any noe i arriving which forms a triangle with j, p the vector x i places a 1/n in the row corresponing to the ege jp. One easily sees again that for the resulting matrix A, any {0,1} solution to Az 2 ientifies a triangle-free subgraph. (An there is an obvious extension for general B.) The rest of the proof is similar to Theorem 2.2 an we omit the etails in this submission. 2.3 Ranomize Online Algorithms It is natural to ask if ranomization can improve the situation for online vector bin packing. We can, however, give similar lower bouns base on results in [14] for ranomize online colouring. They use the same construction as for Theorem 2.1 but appeal to Yao s Lemma [21]. That is, it is enough to exhibit a istribution of k-colourable graphs for which the expecte number of colours use by any eterministic algorithm is at least n/k. As before, the istribution can be chosen to have support restricte to HS graphs of orer k (hence k = O(logn)). The aversary constructs a graph from this istribution in avance an hence oblivious to the choices of the online algorithm. They prove the following for sufficiently large n: the competitive ratio of any ranomize online colouring algorithm is at least least n/(16log 2 n), even if its input is restricte to HS graphs. We may piggyback on their analysis of expecte behaviour of a eterministic algorithm as follows. An HS graph is etermine by a sequence of noes v i together with a subset F(v i ) of amissible colours, where F(v i ) = k/2, an an assignment of colours to the noes (obeying consistency). The istribution is as follows. The graph is generate iteratively where each F(v i ) is a ranom k/2 subset of [k]. Then v i is mae ajacent to the appropriate noes to preserve consistency. It is then assigne a ranom colour from [k] F(v i ) (the amissible colours). Consier {0,1} VBP. We may efine a 1-1 map to the space of n 2 n matrices use in the lower boun of VBP. As we have argue, a solution of size X to the VBP instance immeiately yiels a solution of the same size for the colouring instance. Hence the expecte performance of a eterministic algorithm for VBP coul be no better than that for online colouring. 3 A competitive VBP algorithm In the case of B = 1 our lower bouns are tight. Inee, for the [0,1] case, there is a ( +.7)-competitive algorithm base on first fit [11]. The bouns are also tight for the {0,1} case - see Section 5. We now present an algorithm for B 2. As mention in the introuction, the algorithm is base on two parts. The first part solves V BP using a loa balancing technique with a set of virtual enlarge bins (of size cb log ). The loa balancing uses exponential weights to ecie on the assignment. The secon part assigns vectors of each virtual bin to real various bins of size B. For the secon part we present two algorithms, a ranomize algorithm an a eterministic algorithm. Combining both parts prouces a V BP algorithm with the esire performance. 3.1 The virtual VBP algorithm. In the following, let opt = OPT (x) be the minimum number of bins of size B require to assign vectors sequence x in off-line fashion. We present an online V BP algorithm that uses 4opt virtual bins with size cblog for some constant c > 0. The input is an online stream of vectors x R. Let the i th vector be x i = (x i1,...,x i ) where x ik [0,1] for 1 i n an 1 k. An online loa balancing V BP algorithm 7

8 assigns each vector to a bin. Let A(i) = j if vector i is assigne to bin j by the algorithm. The loa of bin j in coorinate k just before vector i arrives is L i jk = i <i:a(i )= j In the following, when we rop the superscript, then L jk enotes the loas upon termination of the algorithm. Aitionally, let x ik = x ik /B an L i jk = Li jk /B. First we construct a proceure for virtual bin packing for what we call -balance vectors. A vector x i [0,1] is -balance if it satisfies: max r,k,x ik >0 x ir/x ik. Let M i = maxx ik an c 1 > 0,2 > a > 1 be constants that will be etermine later. We now present k Proceure 1 for virtual bin packing of -balance vectors an Algorithm Virtual-VBP that uses Proceure 1 for virtual bin packing of general vectors. m 1. Open one active bin; foreach vector x i o Let j be the active bin with minimal k a L i jk + x ik a L i jk; if k : L i jk + x ik c 1 log then assign vector i to bin j; else f ailure De-activate the m active bins; m 2m. Open an active new m active bins; Assign x i to the first (empty) active bin; en x i k. Proceure 1: Virtual V BP for -balance vectors begin Init Proceure 1 for simulation; foreach vector x i o Define { x i as: x ik = 0 if x ik M i 1. x ik otherwise. Use Proceure 1 to assign vector x i to simulate bin j; Assign x i to a virtual bin j; en en Algorithm Virtual-VBP: Virtual VBP for general vectors Theorem 3.1. Algorithm Virtual-VBP uses at most 4opt bins of size cblog (where c = 2c 1 ). First we establish some properties of Proceure 1 when assigning a sequence x of -balance vectors. 8

9 Lemma 3.2. Proceure 1 never fails on -balance vectors when it uses m opt active bins of size c 1 Blog. We will use Lemma 3.2 to prove the constant (in term of number of bins) boun. Corollary 3.3. When applying Proceure 1 on -balance vectors, Proceure 1 opens at most 4opt bins. of Corollary 3.3. Using Lemma 3.2, if the number of active bins m opt, then Proceure 1 oes not open any new bins. Therefore, the number of active bins is less than 2opt an the total number of opene bins is less than 4opt. In orer to prove Lemma 3.2 we first show the following two lemmas. Lemma 3.4. a > 1 an 0 x 1,a x 1 (a 1)x Proof. The function f (x) = (a 1)x is linear in the section [0,1], but g(x) = a x 1 is convex. The functions intersect at x = 0 an x = 1 an the lemma follows immeiately since the functions are continuous. Lemma 3.5. When applying Proceure 1 on -balance vectors, then for any active bins j, j an for each coorinate k the following hols: a L jk a L j k lna (a 1)a 1/B. Proof. Let i be the last vector assigne to bin j with x i k > 0. Then for any active bin j: The left sie of the inequality satisfies a L i j k + x i k a L i j k a L i jk + x i k a L i jk. a L i j k + x i k a L i j k a L i j k (a x i k 1) a L i j k ln(a) x i k a ( L j k 1/B) ln(a) x i k where the first inequality follows since the coorinate k is part of the sum; the secon inequality follows since for x > 0, e x 1 x (x = x i k ln(a) > 0 for a > 1); the thir inequality follows since i is the last vector assigne to bin j an x i k 1/B. The right sie of the inequality satisfies a L i jk + x i k a L i jk = a L i jk(a x i k 1) a L jk (a 1) x i k. The inequality follows from Lemma 3.4 an monotonicity of a L i jk. From the two parts we get a L jk (a 1) x i k a ( L j k 1/B) x i k lna. 9

10 Since x i is -balance, for all k we have x i k/ x i k an therefore: a L jk a L j k lna (a 1)a 1/B. of Lemma 3.2. Assume there are m opt active bins in Proceure 1. We prove that when assigning all the vectors to the m active bins the loa never excees c 1 Blog an hence there is no failure. Let j opt (i) be the active bin in which the optimal algorithm assigne vector i. We can assume such a bin exists since m opt. By the efinition of Proceure 1 the next chosen bin j satisfies: a L i jk + x ik a L i jk = a L i j opt (i)k + x ik a L i j opt (i)k a L i j opt (i)k(a x ik 1) a L i j opt (i)k(a 1) x ik k a L j opt (i)k (a 1) x ik where the secon inequality follows from Lemma 3.4 an the last inequality follows from the monotonicity of the loa. Summing over all vectors: i By replacing the orer of summation we get a L i j(i)k + x ik a L i j(i)k (a 1) i m j=1 i A(i)= j (a 1) a L j opt (i)k x ik. a L i jk + x ik a L i jk m a L jk j=1 i OPT (i)= j The loa in the optimal assignment is at most B an hence i OPT (i)= j x ik 1. Aitionally, for each j,k the left han sie is a telescoping sum an hence x ik. Therefore, m j=1 m j=1 (a L jk a 0 ) (a 1) a L jk m (a 1) m a L jk. j=1 m a L jk. j=1 10

11 Hence, m j=1 Using Lemma 3.5, for each bin-coorinate pair j,k : By rearranging the terms: a L jk m 2 a. m m a L j k lna (a 1)a 1/B j=1 For fixe a (1,2) we get L j k c 1 log as claime. a L j k 2 (a 1)a1/B (2 a)lna. a L jk m 2 a We are now reay to prove the feasibility an competitiveness of Algorithm Virtual-VBP for general vectors. of Theorem 3.1. Note that the number of bins opene by Algorithm Virtual-VBP on vector sequence x is the same as the number of bins opene by Proceure 1 on vector sequence x. Recall that x ik = x ik if x ik > M i / an 0 otherwise. Therefore, for all i,k we have x ik x ik hence OPT (x ) OPT (x). By Corollary 3.3 the number of bins opene by Proceure 1 is at most 4OPT (X ) 4OPT (X). Aitionally, the loa on each bin is: x ik = (x ik x ik) + x ik i:a(i)= j i:a(i)= j i:a(i)= j i:a(i)= j The secon inequality follows from Lemma 3.2 an from M i + x ik i:a(i)= j c 1Blog + c 1 Blog = 2c 1 Blog. i:a(i)= j M i c 1 Blog (by volume consieration). 3.2 The loa balancing VBP algorithm. In the last section we showe an online algorithm that packs the vectors into virtual bins of size cblog. In orer to solve the V BP we nee to pack into bins with size B. For that, we istribute each virtual bin s vectors into real bins. Initially we open r real bins for each virtual bin an whenever a vector is assigne to this virtual bin, we choose uniformly at ranom one of the real bins associate with it an assign the vector to this real bin if it is feasible. In a case of a failure, i.e. if the loa of the chosen real bin after assignment excees B, then we open r new real bins an irect future vectors to the new real bins. By using Chernoff bouns an choosing r appropriately, the probability of failure is small enough so that the expecte total number of bins associate with each virtual bin is O(r). 11

12 begin Init Algorithm Virtual-VBP for simulation; foreach vector x i o Use Algorithm Virtual-VBP to assign x i to virtual bin j; Use Proceure 2 to assign vector x i of virtual bin j to its real bin; en en Algorithm LB-VBP: The V BP algorithm begin Open r new active real bins; foreach vector x i assigne to this virtual bin o Choose uniformly at ranom active bin j; if k : L i jk + x ik B then Assign vector i to real bin j; else f ailure De-activate the active bins ; Open r new active bins; Assign x i to a ranom active (empty) bin; en en Proceure 2: Distributing vectors from a single virtual bin (ranomize) 12

13 Theorem 3.6. Algorithm LB-VBP, using r = Θ( 1/(B 1) log B/(B 1) ) in Proceure 2, provies a feasible assignment an achieves (expecte) competitive ratio of O( 1/(B 1) log B/(B 1) ). In orer to prove Theorem 3.6 we prove the following lemmas. Lemma 3.7. Given a virtual bin with size cblog. Distribute all of its vectors uniformly at ranom into r = (celog) B/(B 1) (2) 1/(B 1) real bins. The probability that there exists a bin coorinate whose loa excees B is less then 1/2. Proof. Let x 1,x 2,...x N be the vectors assigne to the virtual bin. Let Y j 1,...,Y j N be ranom variables, where Y j i represents whether vector i is irecte to the bin j. Let X jk jk 1,...,XN representing the loa accumulate in bin j at coorinate k by vector i,i.e. X jk i = Y j i x ik. The loa in bin j at coorinate k is X jk = i X jk i. Notice that, E[X jk i ] = x ik /r an E[X jk ] = E[ i X jk i ] = i x ik /r cblog/r B. Let µ k = E[X jk ]. X jk i [0,1] an therefore by a Chernoff boun for any δ k > 1: ( ) e µk δ k Pr[X jk δ k µ k ]. δ k Let δ k = B/µ k 1 an δ = minδ k. Therefore, the probability that overloa occurs in bin j at coorinate k is k ( ) e B ( e ) B Pr[X jk B]. δ k δ Using a union-boun, the probability of an overloa in some bin-pair coorinate is: Pr[ j,k X jk B] rpr[x jk B] ( e ) B r δ ( eclog r r = (eclog)b r B 1 = 1 2. where the last inequality follows from B/δ = max µ k cblog/r an last equality follows from the efinition of r. k Lemma 3.8. Let r = (celog) B/(B 1) (2) 1/(B 1). The expecte number of bins opene by the Proceure 2 is O(r). Proof. By Lemma 3.7 using r = (celog) B/(B 1) (2) 1/(B 1) the probability of opening r aitional bins is less then 1/2. Clearly, the probability of an i th aitional opening of bins given (i 1) aitional openings is less then 1/2. Hence, the probability for opening ir bins is less then 1/2 i. Therefore, the expecte number of open bins is at most 2r. ) B 13

14 of Theorem 3.6. Since Proceure 2 checks that the loa is at most B, the algorithm is feasible. From Lemma 3.1 the number of virtual bins opene is at most 4opt, each with size cblog. From Lemma 3.8 the expecte number of bins opene by each instance for these virtual bin vectors is O( 1/(B 1) log B/(B 1) ). Therefore, the total expecte number of bins is O(opt 1/(B 1) log B/(B 1) ). From Theorem 3.6 we prove the existence of a ranomize algorithm satisfying the first part of Theorem De-ranomizing the loa balancing VBP algorithm. We now present a proceure that istributes the vectors of a single virtual bin eterministically, with up to a constant same number of bins as the ranomize Proceure 2. This yiels a eterministic V BP algorithm with the same competitive ratio as Algorithm LB-VBP. We use e-ranomizing techniques from [1]. Our algorithm woul use T clog bins (where T will be chosen later). We will use potential function, when a vector arrives, the algorithm assigns the vector to one of the bins, if by oing so the following potential function oes not increase. We will show that in each step such a bin exists. Let T = ln(t + 1). Define the potential of coorinate k in bin j after the arrival of vector i as { } Φ i jk = exp T L i i jk x i k clog The potential function after the arrival of vector i is: Φ i = i =1 T clog 1 T clog j=1 Φ i jk begin Open new T c log active bins; foreach vector x i assigne to this virtual bin o Assign vector i to bin j s.t. Φ i Φ i 1. en en Proceure 3: Distributing single virtual bin (e-ranomize) First we prove the correctness of the Proceure 3, we prove that in each step such bin always exists. Lemma 4.1. Initially, Φ 0 1 an throughout the algorithm, Φ i > 0. In aition, when vector i arrives, there exists a bin j such that assigning vector i to bin j oes not increase the potential function. Proof. It is easy to verify that initially the value of the potential function is exactly 1. Also, since the potential function is a sum of exponential functions it is always positive. To prove the secon part of the claim we use a probabilistic argument. When vector i arrives we choose uniformly at ranom one of the bins an assign the vector to it, i.e. each bin is selecte with probability 1/(T clog). We prove that with this ranom trial the expecte value of the potential function oes not increase. This means that there exists a bin j such that assigning the vector i to j oes not increase the potential function. The expecte value of Φ i jk is: 14

15 E[Φ i jk ] = 1 ( ) Φ i 1 jk T clog et x ik x ik clog ( ) 1 ( ) + 1 Φ i 1 x jk T clog e ik clog ( ) = Φ i 1 x jk e ik clog 1 T clog et x ik T clog ( ) = Φ i 1 x jk e ik clog 1 T clog (et x ik 1) + 1 ( ) Φ i 1 x jk e ik clog 1 T clog (et 1)x ik + 1 ( ) = Φ i 1 x jk e ik xik clog clog + 1 Φ i 1 jk where the first inequality from Lemma 3.4 for a = e T an x = x ik, the last equality follows from the efinition of T, the last inequality follows from for any x > 0 we have 1 + x e x. By linearity of expectation we get E[Φ i ] Φ i 1. Lemma 4.2. Given a virtual bin with size cblog. Assigning its vectors using Proceure 3 to T clog bins for T = (ln(clog) + B)/(B 1) (recall that T = e T 1) then the loa on each bin coorinate is at most B. Proof. From Lemma 4.1 the potential Φ 1, since the potential is the summation of positive terms then for all j,k,i we have Φ i jk /(T clog) < 1. Hence By rearranging the terms exp{t L i jk i i =1 x i k } < T clog clog L i jk < ln(t clog) + T ln(t clog) + B T i i =1 T + ln(clog) + B T = B x i k clog where the secon inequality follows from x ik cblog as these are vectors that assigne to a virtual bin, the last inequality implie by the efinition i of T an the last equality implie by the efinition of T. 15

16 To get a eterministic V BP algorithm, we moify Algorithm LB-VBP, by replacing Proceure 2 with Proceure 3. From these two lemmas above we get that this algorithm is feasible an has competitive ratio of O( 1/(B 1) log B/(B 1) ). 5 The VBP algorithm for the {0,1} case. The greey algorithm. For the {0,1} case with B = 1, the greey algorithm is known to give a O( )- competitive algorithm. For completeness we give the analysis an show that it extens also to the B 2 case. The algorithm greeily assigns an arriving vector to a bin with available space. If there is none, we open a new bin. We classify vectors as large if 1 T x i, otherwise small. Let the number of large vectors be k an the number small is n k. Clearly opt k. Now consier a small vector x i that is blocke. Let B 1,B 2,...,B s be the bins opene when it arrive. For each j = 1,2,..., we say x i is j-blocke for bin B p, if x i j = 1 an there is some vector v B p with v j = 1. Let N j be the number of j-blocke bins, then clearly opt N j. Moreover, since x i was small, we have N j > 0 for at most values of j. Since every B p is j-blocke for some j, we have that s N max opt. It follows that the number of bins open at any time is at most 2 opt. Note that this also works for general B. In that case, if there are k large vectors, then opt k B. Now, we say we are j-blocke in a bin if x i j = 1 an that bin alreay contains B vectors with a 1 in component j. If we are j-blocke by N j bins, then again opt N j. Hence s N max opt for an overall boun of 2B opt. Improving the exponent for B 2. For the {0,1} case with B 2, we moify Algorithm LB-VBP by using a ifferent number of real bins in its proceure. Specifically, we use O( 1/B log (B+1)/B ) real bins for each single virtual bin. This results in V BP algorithm for the {0,1} case that has a competitive ratio of O( 1/B log (B+1)/B ). We observe that in the {0,1} case overloa oes not occur if L jk < B + 1. By moifying Lemma 3.7, we prove that istribute a virtual bin vectors uniformly at ranom into r = (celog) (B+1)/B (2) 1/B real bins then the probability that there exists a bin coorinate whose loa excees B + 1 is less then 1/2. For that, we choose δ k = (B + 1)/µ k an we get: Pr[ j,k X jk B + 1] rpr[x jk B + 1] ( e ) B+1 r δ ( ) eclog B+1 r r = (eclog)b+1 r B = 1 2. Now, we can use Proceure 2 with r = O( 1/B log (B+1)/B ). Therefore, the competitive ratio of the moifie Algorithm LB-VBP for the {0,1} case is O( 1/B log (B+1)/B ). Accoringly, we can get a eterministic algorithm with the same competitive ratio by using T = (ln(clog) + B)/B in Proceure 3. The choice of this T in Lemma 4.2 implies that L i jk < B+1, which prove feasibility for {0,1} vectors an yiels the same competitive ratio. 16

17 6 Open Questions The most interesting open problem is to close the small but intriguing gap between the upper an lower bouns for general bins B 2. In particular for the case B = Θ(log), there is still a gap between a logarithmic upper boun to a constant lower boun. Another interesting question arises from the filtering algorithm employe in the proof of Theorem 2.2 for online colouring K 3 -free HS graphs. Namely, that algorithm neee the structure of HS graphs. Can this be eliminate? Concretely, is there an online colouring algorithm for the class of K 3 -free graphs whose competitive ratio is O(n 1 β ) for some β > 0? While such graphs are n-colourable, it appears pruent (necessary?) to use K 3 -freeness more irectly. This is suggeste by the fact that there is no n 1 β -competitive algorithm for colouring k-colourable graphs [16, 17] (inee there is barely a sublinear boun). One approach might be to leverage the polylog competitive algorithm of [17] for colouring bipartite graphs. There are natural extensions of the question to the class of K B -free graphs for larger B. Finally, the upper bouns are not quite tight in their exponents; this remains a challenge to explore. 7 Acknowlegements This work was partly performe while the first an thir authors were visiting Microsoft Research s Theory Group uring the summer of The thir author also acknowleges the kin hospitality of his host James R. Lee while visiting the CSE Department at the University of Washington uring 2012, an acknowleges support from the NSERC Discovery an Accelerator Programmes. A Easy lower boun argument for Online Packing Version The lower boun for online stable set mentione in Section 1.2 can be aapte easily to online PIP as follows. The aversary fees a sequence of noes (vectors incluing all past an future ajacencies) v 1,v 2,...v i each of which is ajacent to noes v n/2+1,v n/2+2,...,v n until either (a) we reach i = n/2 or (b) the algorithm selects some v i. In Case (a), the aversary then fees the remaining noes v j : j > n/2 to form a clique. Hence the algorithm misse the n/2 stable set v j : j n/2, an can prouce at best a stable set of size 1 here on in. In Case (b), the aversary will fee v i+1,...,v n/2 as a clique so the algorithm may pick at most one of these. The aversary then fees v j : j > n/2 as a stable set, but then the algorithm cannot pick any of these as it committe to v i. References [1] N. Buchbiner an J. Naor. Online primal-ual algorithms for covering an packing problems. 13th Annual European Symposium on Algorithms - ESA 2005, [2] C. Chekuri an S. Khanna. On multi-imensional packing problems. SIAM journal on computing, 33(4): , [3] E.G. Coffman Jr, M.R. Garey, an D.S. Johnson. Approximation algorithms for bin packing: A survey. In Approximation algorithms for NP-har problems, pages PWS Publishing Co.,

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