Packing Small Vectors

Azar, Yossi; Cohen, Ilan Reuven; Fiat, Amos; Roytman, Alan

doi:10.1137/1.9781611974331.ch103

Cited by 20 publications

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“…The solution is feasible as the size of any second phase item is in (1 − 3 −2 2N+2 , 1 − 3 −2 2N+3 ), any two first phase items have a total size of at most 2 · 3 −2 2N+4 < 3 −2 2N+3 , and those second phase items packed with third phase items have sizes no larger than 1 − ζ 2 . This solution is optimal since its cost is equal to the number of second phase items, and their sizes are above 1 2 . Thus, OP T (J 4 ) − 1 ≤ d + N 2 .…”

Section: K =mentioning

confidence: 99%

“…The previously known lower bounds for vector packing are as follows. The best results for constant dimensions are fairly low, and tending to 2 as the dimension d grows to infinity [12,7,6], while a lower bound of Ω(d 1−ε ) was given by Azar et al [2] for the case where both d and the optimal cost are functions of a common parameter n that grow to infinity when n grows to infinity, and thus this result does not give any lower bound on the competitive ratio for constant values of d (see also [1] for results on vectors with small components). In particular, the best lower bound for d = 2 prior to this work was 1.67117 [12,7,6].…”

Section: Introductionmentioning

confidence: 99%

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Online bin packing with cardinality constraints resolved

Balogh

Békési

Dósa

et al. 2020

Journal of Computer and System Sciences

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Cardinality constrained bin packing or bin packing with cardinality constraints is a basic bin packing problem. In the online version with the parameter k ≥ 2, items having sizes in (0, 1] associated with them are presented one by one to be packed into unit capacity bins, such that the capacities of bins are not exceeded, and no bin receives more than k items. We resolve the online problem in the sense that we prove a lower bound of 2 on the overall asymptotic competitive ratio. This closes this long standing open problem, since an algorithm of an absolute competitive ratio 2 is known. Additionally, we significantly improve the known lower bounds on the asymptotic competitive ratio for every specific value of k. The novelty of our constructions is based on full adaptivity that creates large gaps between item sizes. Thus, our lower bound inputs do not follow the common practice for online bin packing problems of having a known in advance input consisting of batches for which the algorithm needs to be competitive on every prefix of the input.

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Section: K =mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Online bin packing with cardinality constraints resolved

Balogh

Békési

Dósa

et al. 2020

Journal of Computer and System Sciences

View full text Add to dashboard Cite

show abstract

“…To overcome the strong lower bound for arbitrary d, Epstein [18] initiated the study of VBP with variable bin sizes. In the online setting, the O(d) bound was improved tõ O(d vectors are splittable, [5] gave an e-competitive algorithm, and a matching lower bound was shown in [7].…”

Section: Online Vector Bin Packing (Vbp)mentioning

confidence: 99%

“…On the other hand, if the number of machines is fixed, and the goal is to minimize the maximum load on any machine, then the problem is called list scheduling (or online makespan minimization) and has been studied since the work of Graham in the 60s (see, e.g., [23,24] and surveys [3,33,2] for later work). Much of the recent interest in these problems has been for the vector bin packing (VBP) and vector scheduling (VS) problems, where jobs have vector loads and the goal is to balance the load on all dimensions simultaneously [6,5,28,25]. This is motivated by applications such as data center scheduling, where each job requires multiple resources (each resource is a dimension) for its execution.…”

Section: Introductionmentioning

confidence: 99%

Randomized Algorithms for Online Vector Load Balancing

Azar¹,

Cohen²,

Panigrahi³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study randomized algorithms for the online vector bin packing and vector scheduling problems. For vector bin packing, we achieve a competitive ratio ofÕ(d 1/B ), where d is the number of dimensions and B the size of a bin. This improves the previous bound ofÕ(d 1/(B−1) ) by a polynomial factor, and is tight up to logarithmic factors. For vector scheduling, we show a lower bound of Ω( log d log log d ) on the competitive ratio of randomized algorithms, which is the first result for randomized algorithms and is asymptotically tight. Finally, we analyze the widely used "power of two choices' algorithm for vector scheduling, and show that its competitive ratio is O(log log n + log d log log d ), which is optimal up to the additive O(log log n) term that also appears in the scalar version of this algorithm.

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“…The multidimensional version of this problem was introduced by Chekuri and Khanna in the offline model [13], who gave a PTAS for constant d. For unrelated machines, they showed a constant lower bound, and the best known approximation factor is O(log d/ log log d) due to Harris and Srinivasan [21]. In the online setting, Azar et al [6] and Meyerson et al [28] In the next section, we present the idea of machine smoothing that is a generic tool we use in all the algorithms. This is essentially sufficient for minimizing makespan in vector scheduling on homogeneous machines (Section 3), but we need more ideas for minimizing arbitrary q-norms.…”

Section: (Section 4 and Section 5)mentioning

confidence: 99%

Online load balancing on related machines

Kell

Panigrahi

et al. 2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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In the load balancing (or job scheduling) problem, introduced by Graham in the 1960s (SIAM J. of Appl. Math. 1966, 1969, jobs arriving online have to be assigned to machines so to minimize an objective defined on machine loads. A long line of work has addressed this problem for both the makespan norm and arbitrary ℓ q -norms of machine loads. Recent literature (e.g., Azar et al., STOC 2013; Im et al., FOCS 2015) has further expanded the scope of this problem to vector loads, to capture jobs with multi-dimensional resource requirements in applications such as data centers. In this paper, we completely resolve the job scheduling problem for both scalar and vector jobs on related machines, i.e., where each machine has a given speed and the time taken to process a job is inversely proportional to the speed of the machine it is assigned on. We show the following results:• Scalar scheduling. We give a constant competitive algorithm for optimizing any ℓ q -norm for (scalar) scheduling on related machines. The only previously known result was for the makespan norm. *The load balancing (or job scheduling) problem, introduced in the seminal work of Graham in the 1960s [19,20], asks for an online assignment of jobs to machines so as to minimize some objective defined on machine loads. A long line of work has addressed this problem for both the makespan norm (maximum load) and for other ℓ q -norms of machine loads (e.g., [9, 26, 2, 16, 2, 15, 10, 18, 22, 3, 8, 11, ?, 4, 12]). In this paper, we study this problem in the related machines setting, where the processing time of a job on a machine is inversely proportional to the speed of the machine. The only previous result for this problem on related machines was a constant-competitive algorithm for the makespan (maximum load) objective [11]. However, in many situations, other ℓ q -norms of machine loads are more relevant: e.g., the 2-norm is suitable for disk storage [?, ?], whereas q between 2 and 3 is used for modeling energy consumption [?, ?, ?]. This led to constant-competitive algorithms for arbitrary ℓ q -norms of machine loads for the special case of identical machines (all machine speeds are equal) [?], and to O(q)-competitive algorithms for the more general unrelated machines setting (processing times are arbitrary) [4,12]. But, this problem has remained open for related machines. Moreover, recent literature has further expanded the scope of the job scheduling problem to vector jobs that have multiple dimensions, the resulting problem being called vector scheduling [13,7,28,25]. This problem is very relevant to scheduling on data centers where jobs with multiple resource requirements have to be allocated to machine clusters to make efficient use of limited resources such as CPU, memory, network bandwidth, and storage [17,29,27,14,24,25]. Recently, Im et al. [23] showed that for vector scheduling with the makespan norm, competitive ratios of O(log d/ log log d) and O(log d + log m) are tight for identical and unrelated machines respectively, where d is the number o...

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Packing Small Vectors

Cited by 20 publications

References 0 publications

Online bin packing with cardinality constraints resolved

Online bin packing with cardinality constraints resolved

Randomized Algorithms for Online Vector Load Balancing

Online load balancing on related machines

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