On-line scheduling of multi-core processor tasks with virtualization

We consider an on-line list scheduling problem of multi-core processor tasks with virtualization to minimize makespan. The competitive ratio of an on-line algorithm is shown for every specific m, where m is the number of processors. Better on-line algorithms are presented for a small number of processors.     

Parallel searching on m rays

We investigate parallel searching on m concurrent rays. We assume that a target t is located somewhere on one of the rays; we are given a group of m point robots each of which has to reach t. Furthermore, we assume that the robots have no way of communicating over distance. Given a strategy S we are interested in the competitive ratio defined as the ratio of the time needed by the robots to reach t using S and the time needed to reach t if the location of t is known in advance.

If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of 9—independent of m. We show that 9 is a lower bound on the competitive ratio for two large classes of strategies if m2.

If the minimum distance to the target is not known in advance, we show a lower bound on the competitive ratio of 1+2(k+1)^k+1/k^k where k=logm where log is used to denote the base-2 logarithm. We also give a strategy that obtains this ratio.     

On-line arbitrarily vertex decomposable trees

A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by τ. An on-line version of the problem of characterizing arbitrarily vertex decomposable trees is completely solved here.     

A Review of On-Line Machine Scheduling:Algorithms and Competitiveness

在过去的十年里，在线算法的研究吸引了广泛的兴趣．本文对在排序和时间表问题中的各种有效的在线算法以及它们的竞争度作一综述．     

Online weighted flow time and deadline scheduling

In this paper we study some aspects of weighted flow time. We first show that the online algorithm Highest Density First is an O(1)-speed O(1)-approximation algorithm for P|r_i,pmtn|∑w_iF_i. We then consider a related Deadline Scheduling Problem that involves minimizing the weight of the jobs unfinished by some unknown deadline D on a uniprocessor. We show that any c-competitive online algorithm for weighted flow time must also be c-competitive for deadline scheduling. We then give an O(1)-competitive algorithm for deadline scheduling.     

On-line Chain Partitioning of Up-growing Interval Orders

On-line chain partitioning problem of on-line posets has been open for the past 20 years. The best known on-line algorithm uses chains to cover poset of width w. Felsner (Theor. Comput. Sci., 175(2):283–292, 1997) introduced a variant of this problem considering only up-growing posets, i.e. on-line posets in which each new point is maximal at the moment of its arrival. He presented an algorithm using chains for width w posets and proved that his solution is optimal. In this paper, we study on-line chain partitioning of up-growing interval orders. We prove lower bound and upper bound to be 2w−1 for width w posets. Piotr Micek and Bartłomiej Bosek are scholars of the project which is co-financed from the European Social Fund and national budget in the frame of The Integrated Regional Operational Programme.     

Semi-on-line scheduling problems for maximizing the minimum machine completion time

1. IntroductionThis paper considers the following scheduling problem: We are given a set J = {pl,Pz,', p.} of independent jobs, each with a positive processing time pi, that must be scheduled on m > 1 parallel and identical machines M = {MI, M2,', Mm}. We identify thejobs with their processing times. The jobs and machines are available at time zero, andno preemption is allowed. The objective is to maximize the minimum workload over themaChines. This problem, denoted by PtICmi., is on…     

P‖Cmin随机算法研究

本文研究了P‖Cmin的随机算法及其最坏情况界，我们给出了Pm‖Cmin在线排序问题新的随机上界，并给出了P2‖Cmin的最好随机算法，其最坏情况界为2／3。对P2‖Cmin已知工件加工时间递减半在线模型，我们给出了一最坏情况界为6／7的随机算法并证明了它为最好的。     

On-line algorithms for networks of temporal constraints

We consider a semi-dynamic setting for the Temporal Constraint Satisfaction Problem (TCSP), where we are requested to maintain the path-consistency of a network under a sequence of insertions of new (further) constraints between pairs of variables. We show how to maintain the path-consistency in O(nR³) amortized time on a sequence of Θ(n²) insertions, where n is the number of vertices of the network and R is its range, defined as the maximum size of the minimum interval containing all the intervals of a single constraint.Furthermore we extend our algorithms to deal with more general temporal networks where variables can be points and/or intervals and constraints can also be defined on pairs of different kinds of variables. For such cases our algorithms maintain their performance. Finally, we adapt our algorithms to also maintain the arc-consistency of such generalized networks in O(R) amortized time for Θ(n²) insertions.     

On-line scheduling with non-crossing constraints

We consider the problem of on-line scheduling with non-crossing constraints. The objective is to minimize the latest completion time. We provide optimal competitive ratio heuristics for the on-line list and on-line time problems with unit processing times, and a 3-competitive heuristic for the general on-line time problem.     

On-Line Potato-Sack Algorithm Efficient for Packing Into Small Boxes

The aim of this paper is to show an on-line algorithm for packing sequences of d-dimensional boxes of edge lengths at most 1 in a box of edge lengths at least 1. It is more efficient than a previously known algorithm in the case when packing into a box with short edges. In particular, our method permits packing every sequence of boxes of edge lengths at most 1 and of total volume at most in the unit cube. For packing sequences of convex bodies of diameters at most 1 the result is d! times smaller.     

Randomized On-Line Scheduling Similar Jobs to Minimize Makespan on Two Identical Processors

In this paper we consider an on-line scheduling problem, where jobs with similar processing times within [1, r] arrive one by one to be scheduled in an on-line setting on two identical parallel processors without preemption. The objective is to nlinimize makespan. We devise a randomized on-line algorithm for this problem along with a lower bound.     

On-line scheduling on a batch processing machine with unbounded batch size to minimize the makespan

We present on-line algorithms to minimize the makespan on a single batch processing machine. We consider a parallel batching machine that can process up to b jobs simultaneously. Jobs in the same batch complete at the same time. Such a model of a batch processing machine has been motivated by burn-in ovens in final testing stage of semiconductor manufacturing. We deal with the on-line scheduling problem when jobs arrive over time. We consider a set of independent jobs. Their number is not known in advance. Each job is available at its release date and its processing requirement is not known in advance. This general problem with infinite machine capacity is noted 1∣p − batch, r_j, b = ∞∣C_max. Deterministic algorithms that do not insert idle-times in the schedule cannot be better than 2-competitive and a simple rule based on LPT achieved this bound [Z. Liu, W. Yu, Scheduling one batch processor subject to job release dates, Discrete Applied Mathematics 105 (2000) 129–136]. If we are allowed to postpone start of jobs, the performance guarantee can be improved to 1.618. We provide a simpler proof of this best known lower bound for bounded and unbounded batch sizes. We then present deterministic algorithms that are best possible for the problem with unbounded batch size (i.e., b = ∞) and agreeable processing times (i.e., there cannot exist an on-line algorithm with a better performance guarantee). We then propose another algorithm that leads to a best possible algorithm for the general problem with unbounded batch size. This algorithm improves the best known on-line algorithm (i.e. [G. Zhang, X. Cai, C.K. Wong, On-line algorithms for minimizing makespan on batch processing machines, Naval Research Logistics 48 (2001) 241–258]) in the sense that it produces a shortest makespan while ensuring the same worst-case performance guarantee.     

Scheduling uniform machines on-line requires nondecreasing speed ratios

We consider the following on-line scheduling problem. We have to schedulen independent jobs, wheren is unknown, onm uniform parallel machines so as to minimize the makespan; preemption is allowed. Each job becomes available at its release date, and this release date is not known beforehand; its processing requirement becomes known at its arrival. We show that if only a finite number of preemptions is allowed, there exists an algorithm that solves the problem if and only ifs _i–1/s_i s_i/s_i+1 for alli = 2,,m – 1, wheres _i denotes theith largest machine speed. We also show that if this condition is satisfied, then O(mn) preemptions are necessary, and we provide an example to show that this bound is tight. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

On the effectiveness of the Harmonic Shelf Algorithm for on-line strip packing

In [J. Csirik, G.J. Woeginger, An on-line algorithm for multidimensional bin packing, Inform. Process. Lett. 63 (1997) 171-175] the authors study the asymptotic worst case ratio between the height of the strip needed to on-line pack a list of boxes by means of the Harmonic Shelf Algorithm and the height of the strip used by an optimal algorithm. In this note we analyze the effectiveness of the former algorithm in terms of the ratio between the unused area inside the strip and the total size of this strip, and we show that the Harmonic Shelf Algorithm is also capable of packing items so that the asymptotic worst case value of this ratio comes arbitrarily close to .     

带机器准备时间的平行机在线与半在线排序

本文研究带机器准备时间的m台平行机系统在线和半在线排序问题.对在线排序问题,我们证明了LS算法的最坏情况界为2-1/m.对已知工件加工时间递减,已知总加工时间和已知工件最大加工时间三个半在线模型,我们分析了它们的下界和所给算法的最坏情况界.对其中两台机情形均得到了最好近似算怯。     

局内经济决策问题的竞争分析--一种新的经济数学工具

基于优化领域的热点研究方向之一的局内问题与竞争策略理论,本文提出了局内经济决策问题的一系列概念,说明了处理局内经济决策问题的竞争策略和传统方法的区别以及后者的缺陷.构建了利用局内问题及其竞争策略研究局内经济决策问题的理论框架,并介绍了一个具体研究实例.     

Optimal on-line flow time with resource augmentation

We study the problem of scheduling n jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has ? machines. We design an algorithm of competitive ratio , where Δ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant ?. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only Δ and n are known. This gives a trade-off between the resource augmentation and the competitive ratio.We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has m machines and the on-line algorithm has ?m machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines, and scheduling so as to minimize the total completion time.     

一类带多重核元集在线装箱问题

本文讨论了一类在线变尺寸装箱问题,假定箱子的尺寸可以是不同的.箱子是在线到达的,仅当箱子到达后其尺寸才知道.给定一个带有核元的物品表及其上的核元关系图.我们的目标是要将表中元素装入到达的箱子中,保证任何箱子所装物品不互为核元,即所装物品对应的点所导出的子图是个空图,并使得所用的箱子总长最小.我们证明了该问题是NPHard的,并给出了基于图的点染色、图的团分解和基于背包问题的近似算法,给出了算法的时间复杂度和性能界.     

On-line scheduling of unit time jobs with rejection: minimizing the total completion time

We consider on-line scheduling of unit time jobs on a single machine with job-dependent penalties. The jobs arrive on-line (one by one) and can be either accepted and scheduled, or be rejected at the cost of a penalty. The objective is to minimize the total completion time of the accepted jobs plus the sum of the penalties of the rejected jobs.We give an on-line algorithm for this problem with competitive ratio . Moreover, we prove that there does not exist an on-line algorithm with competitive ratio better than 1.63784.