Bin packing approximation algorithms: Survey and classification

Data center applications present significant opportunities for multiplexing server resources. Virtualization technology makes it easy to move running application across physical machines. In this paper, we present an approach that uses virtualization technology to allocate data center resources dynamically based on application demands and support green computing by optimizing the number of servers actively used. We abstract this as a variant of the relaxed on-line bin packing problem and develop a practical, efficient algorithm that works well in a real system. We adjust the resources available to each VM both within and across physical servers. Extensive simulation and experiment results demonstrate that our system achieves good performance compared to the existing work.

Adaptive Resource Provisioning for the Cloud Using Online Bin Packing

We study scheduling problems in battery-operated computing devices, aiming at schedules with low total energy consumption. While most of the previous work has focused on finding feasible schedules in deadline-based settings, in this paper we are interested in schedules that guarantee good response times. More specifically, our goal is to schedule a sequence of jobs on a variable speed processor so as to minimize the total cost consisting of the power consumption and the total flow time of all the jobs. We first show that when the amount of work, for any job, may take an arbitrary value, then no online algorithm can achieve a constant competitive ratio. Therefore, most of the paper is concerned with unit-size jobs. We devise a deterministic constant competitive online algorithm and show that the offline problem can be solved in polynomial time.

Energy-efficient algorithms for flow time minimization

In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [LMM03], and no approximation better than 1 4 is possible by any algorithm that examines equilibria involving fewer than log n strategies [Alt94]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a 1 2-approximate Nash equilibrium in any 2-player game. For the more demanding notion of well-supported approximate equilibrium due to [DGP06] no nontrivial bound is known; we show that the problem can be reduced to the case of win-lose games (games with all utilities 0-1), and that an approximation of 5 6 is possible contingent upon a graph-theoretic conjecture.

A note on approximate nash equilibria

We study an energy conservation problem where a variable-speed processor is equipped with a sleep state. Executing jobs at high speeds and then setting the processor asleep is an approach that can lead to further energy savings compared to standard dynamic speed scaling. We consider classical deadline-based scheduling, i.e. each job is specified by a release time, a deadline and a processing volume. For general convex power functions, Irani et al. [12] devised an offline 2-approximation algorithm. Roughly speaking, the algorithm schedules jobs at a critical speed Scrit that yields the smallest energy consumption while jobs are processed. For power functions P(s) = sα + γ, where s is the processor speed, Han et al. [11] gave an (αα + 2)-competitive online algorithm.We investigate the offline setting of speed scaling with a sleep state. First we prove NP-hardness of the optimization problem. Additionally, we develop lower bounds, for general convex power functions: No algorithm that constructs Scrit-schedules, which execute jobs at speeds of at least scrit, can achieve an approximation factor smaller than 2. Furthermore, no algorithm that minimizes the energy expended for processing jobs can attain an approximation ratio smaller than 2.We then present an algorithmic framework for designing good approximation algorithms. For general convex power functions, we derive an approximation factor of 4/3. For power functions P(s) = βsα + γ, we obtain an approximation of 137/117

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973099.100

Race to idle: new algorithms for speed scaling with a sleep state

This article extends the study of online algorithms for energy-efficient deadline scheduling to the overloaded setting. Specifically, we consider a processor that can vary its speed between 0 and a maximum speed T to minimize its energy usage (the rate is believed to be a cubic function of the speed). As the speed is upper bounded, the processor may be overloaded with jobs and no scheduling algorithms can guarantee to meet the deadlines of all jobs. An optimal schedule is expected to maximize the throughput, and furthermore, its energy usage should be the smallest among all schedules that achieve the maximum throughput. In designing a scheduling algorithm, one has to face the dilemma of selecting more jobs and being conservative in energy usage. If we ignore energy usage, the best possible online algorithm is 4-competitive on throughput [Koren and Shasha 1995]. On the other hand, existing work on energy-efficient scheduling focuses on a setting where the processor speed is unbounded and the concern is on minimizing the energy to complete all jobs; O(1)-competitive online algorithms with respect to energy usage have been known [Yao et al. 1995; Bansal et al. 2007a; Li et al. 2006]. This article presents the first online algorithm for the more realistic setting where processor speed is bounded and the system may be overloaded; the algorithm is O(1)-competitive on both throughput and energy usage. If the maximum speed of the online scheduler is relaxed slightly to (1+ϵ)T for some ϵ > 0, we can improve the competitive ratio on throughput to arbitrarily close to one, while maintaining O(1)-competitiveness on energy usage.

/pdf/optimizing-throughput-and-energy-in-online-deadline-3dx4qwcanf.pdf

Optimizing throughput and energy in online deadline scheduling

This paper studies the dynamic bin packing problem, in which items arrive and depart at arbitrary times. We want to pack a sequence of unit fractions items (i.e., items with sizes 1/w for some integer w>=1) into unit-size bins, such that the maximum number of bins ever used over all time is minimized. Tight and almost-tight performance bounds are found for the family of any-fit algorithms, including first-fit, best-fit, and worst-fit. In particular, we show that the competitive ratio of best-fit and worst-fit is 3, which is tight, and the competitive ratio of first-fit lies between 2.45 and 2.4942. We also show that no on-line algorithm is better than 2.428-competitive.

/pdf/dynamic-bin-packing-of-unit-fractions-items-1adgeu8ds8.pdf

Dynamic bin packing of unit fractions items

We study the dynamic bin packing problem introduced by Coffman, Garey and Johnson. This problem is a generalization of the bin packing problem in which items may arrive and depart from the packing dynamically. The main result in this paper is a lower bound of 2.5 on the achievable competitive ratio, improving the best known 2.428 lower bound, and revealing that packing items of restricted form like unit fractions (i.e., of size 1/k for some integer k), for which a 2.4985-competitive algorithm is known, is indeed easier.

We also investigate the resource augmentation version of the problem where the on-line algorithm can use bins of size b (>1) times that of the optimal off-line algorithm. An interesting result is that we prove b=2 is both necessary and sufficient for the on-line algorithm to match the performance of the optimal off-line algorithm, i.e., achieve 1-competitiveness. Further analysis gives a trade-off between the bin size multiplier 1<b≤2 and the achievable competitive ratio.

/pdf/on-dynamic-bin-packing-an-improved-lower-bound-and-resource-qq2h4c2a1y.pdf

On Dynamic Bin Packing: An Improved Lower Bound and Resource Augmentation Analysis

Given a mobile telephone network, whose geographical coverage area is divided into cells, phone calls are serviced by assigning frequencies to them, so that no two calls emanating from the same or neighboring cells are assigned the same frequency Assuming an online arrival of calls and the calls will not terminate, the problem is to minimize the span of frequencies usedBy first considering χ-colorable networks, which is a generalization of (the 3-colorable) cellular networks, we present a (χ+1)/2-competitive online algorithm This algorithm, when applied to cellular networks, is effectively a positive solution to the open problem posed in [8]: Does a 2-competitive online algorithm exist for frequency allocation in cellular networks? We further prove a lower bound which shows that our 2-competitive algorithm is optimalWe discover that an interesting phenomenon occurs for the online frequency allocation problem when the number of calls considered becomes large: previously-derived optimal (lower and upper) bounds on competitive ratios no longer hold true For cellular networks, we show new asymptotic lower and upper bounds of 15 and 19126, respectively, which breaks through the optimal bound of 2 shown previously

/pdf/online-frequency-allocation-in-cellular-networks-485t567ff5.pdf

Online frequency allocation in cellular networks

We study the online frequency allocation problem for wireless linear (highway) cellular networks, where the geographical coverage area is divided into cells aligned in a line Calls arrive over time and are served by assigning frequencies to them, and no two calls emanating from the same cell or neighboring cells are assigned the same frequency The objective is to minimize the span of frequencies used

In this paper we consider the problem with or without the assumption that calls have infinite duration If there is the assumption, we propose an algorithm with absolute competitive ratio of 3/2 and asymptotic competitive ratio of 1382 The lower bounds are also given: the absolute one is 3/2 and the asymptotic one is 4/3 Thus, our algorithm with absolute ratio of 3/2 is best possible We also prove that the Greedy algorithm is 3/2-competitive in both the absolute and asymptotic cases For the problem without the assumption, ie calls may terminate at arbitrary time, we give the lower bounds for the competitive ratios: the absolute one is 5/3 and the asymptotic one is 14/9 We propose an optimal online algorithm with both competitive ratio of 5/3, which is better than the Greedy algorithm, with both competitive ratios 2

Joseph Wun-Tat Chan

Papers

Optimizing throughput and energy in online deadline scheduling

Dynamic bin packing of unit fractions items

On Dynamic Bin Packing: An Improved Lower Bound and Resource Augmentation Analysis

Online frequency allocation in cellular networks

Frequency allocation problems for linear cellular networks