A note on the lower bound for online strip packing

This note presents a lower bound of $3/2+\sqrt{33}/6 \approx 2.457$ on the competitive ratio for online strip packing. The instance construction we use to obtain the lower bound was first coined by Brown, Baker and Katseff (1980). Recently this instance construction is used to improve the lower bound in computer aided proofs. We derive the best possible lower bound that can be obtained with this instance construction

Аналіз генетичних алгоритмів розв’язання задачі двовимірної ортогональної упаковки прямокутних об’єктів у напівнескінченну смугу

Досліджено клас генетичних алгоритмів вирішення задачі двовимірної ортогональної упаковки прямокутних об’єктів у напівнескінченну смугу фіксованої ширини. Наведено результати теоретичного аналізу складності реалізації декодерів MERA та BLF; запропоновані власні реалізації цих декодерів з низкою евристичних оптимізацій. Запропоновано реалізацію генетичного алгоритму розв’язання задачі упаковки для окремих випадків (із забороною поворотів об’єктів та з поворотами на 90 °). Описано результати тестових випробувань розробленого алгоритму за різних конфігурацій основних параметрів з використанням загальновідомих тестових наборів. Наведено результати порівняння отриманого алгоритму з іншими відомими алгоритмами.Исследован класс генетических алгоритмов решения задачи двухмерной ортогональной упаковки прямоугольных объектов в полубесконечную полосу фиксированной ширины. Приведены результаты теоретического анализа сложности реализации декодеров MERA и BLF; предложены собственные реализации этих декодеров с рядом эвристических оптимизаций. Предложена реализация генетического алгоритма решения задачи упаковки для отдельных случаев (с запретом поворотов объектов и с поворотами на 90°). Описаны результаты тестирования разработанного алгоритма при разных конфигурациях основных параметров с использованием общеизвестных тестовых наборов. Приведены результаты сравнения полученного алгоритма с другими известными алгоритмами.A class of genetic algorithms for solving the 2D Strip Packing Problem is investigated. The theoretical analysis of the complexity of implementing decoders MERA and BLF is done. Original implementations of these MERA and BLF decoders enhanced with a number of heuristic optimizations are proposed. Genetic algorithm for solving the 2D Strip Packing Problem for special cases (allowed/forbidden objects rotation by 90°) with the use of MERA/BLF decoders is proposed. Extensive computational experiments with well-known instances are performed to analyze different configurations of basic parameters of proposed genetic algorithm. The comparison of the obtained algorithm with other known algorithms is given

Reduction of connections for multibus organization

The multibus interconnection network is an attractive solution for connecting processors and memory modules in a multiprocessor with shared memory. It provides a throughput which is intermediate between the single bus and the crossbar, with a corresponding intermediate cost.Postprint (published version

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition

Scheduling of data-intensive workloads in a brokered virtualized environment

Providing performance predictability guarantees is increasingly important in cloud platforms, especially for data-intensive applications, for which performance depends greatly on the available rates of data transfer between the various computing/storage hosts underlying the virtualized resources assigned to the application. With the increased prevalence of brokerage services in cloud platforms, there is a need for resource management solutions that consider the brokered nature of these workloads, as well as the special demands of their intra-dependent components. In this paper, we present an offline mechanism for scheduling batches of brokered data-intensive workloads, which can be extended to an online setting. The objective of the mechanism is to decide on a packing of the workloads in a batch that minimizes the broker's incurred costs, Moreover, considering the brokered nature of such workloads, we define a payment model that provides incentives to these workloads to be scheduled as part of a batch, which we analyze theoretically. Finally, we evaluate the proposed scheduling algorithm, and exemplify the fairness of the payment model in practical settings via trace-based experiments

Graph Packing over a Rooted Tree

This paper investigates the computational complexity of the graph packing problem over a rooted tree (GPT) as a generalization of the one dimensional bin packing problem, where both the bins and the set of items to be packed are rooted trees. GPT is defined under two problem settings, edge GPT (EPT) and node GPT (NPT). In EPT, the items packed in a bin cannot share any edge but can share some node, while in NPT, the items can share neither node nor edge. We first prove that these problems are in general NP-complete, which strongly suggests that these problems are computationally intractable. However, for the case where the number k of different kinds of items is fixed, we derive a recursive formula of dynamic programming for the minimum number of bins required to pack all the items. This formula can be solved in polynomial time, if the bins and items are all uniform trees and/or comb-shaped trees in which each non-leaf node has the same number of sons. Furthermore, for GPT's with bins of uniform (d, H) trees and only one kind of item, of uniform (d, h) trees, we derive explicit formulas for the number of bins required