17,680 research outputs found

    Two semi-online scheduling problems on two uniform machines

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    Author name used in this publication: C. T. NgAuthor name used in this publication: T. C. E. Cheng2008-2009 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

    New Results on Online Resource Minimization

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    We consider the online resource minimization problem in which jobs with hard deadlines arrive online over time at their release dates. The task is to determine a feasible schedule on a minimum number of machines. We rigorously study this problem and derive various algorithms with small constant competitive ratios for interesting restricted problem variants. As the most important special case, we consider scheduling jobs with agreeable deadlines. We provide the first constant ratio competitive algorithm for the non-preemptive setting, which is of particular interest with regard to the known strong lower bound of n for the general problem. For the preemptive setting, we show that the natural algorithm LLF achieves a constant ratio for agreeable jobs, while for general jobs it has a lower bound of Omega(n^(1/3)). We also give an O(log n)-competitive algorithm for the general preemptive problem, which improves upon the known O(p_max/p_min)-competitive algorithm. Our algorithm maintains a dynamic partition of the job set into loose and tight jobs and schedules each (temporal) subset individually on separate sets of machines. The key is a characterization of how the decrease in the relative laxity of jobs influences the optimum number of machines. To achieve this we derive a compact expression of the optimum value, which might be of independent interest. We complement the general algorithmic result by showing lower bounds that rule out that other known algorithms may yield a similar performance guarantee

    Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

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    In the paper we consider the problem of scheduling nn identical jobs on 4 uniform machines with speeds s1s2s3s4,s_1 \geq s_2 \geq s_3 \geq s_4, respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree Δ\Delta, where two incompatible jobs cannot be processed on the same machine. We show that the problem is NP-hard even if s1=s2=s3s_1=s_2=s_3. If, however, Δ4\Delta \leq 4 and s112s2s_1 \geq 12 s_2, s2=s3=s4s_2=s_3=s_4, then the problem can be solved to optimality in time O(n1.5)O(n^{1.5}). The same algorithm returns a solution of value at most 2 times optimal provided that s12s2s_1 \geq 2s_2. Finally, we study the case s1s2s3=s4s_1 \geq s_2 \geq s_3=s_4 and give an O(n1.5)O(n^{1.5})-time 32/1532/15-approximation algorithm in all such situations

    Probabilistic alternatives for competitive analysis

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    In the last 20 years competitive analysis has become the main tool for analyzing the quality of online algorithms. Despite of this, competitive analysis has also been criticized: it sometimes cannot discriminate between algorithms that exhibit significantly different empirical behavior or it even favors an algorithm that is worse from an empirical point of view. Therefore, there have been several approaches to circumvent these drawbacks. In this survey, we discuss probabilistic alternatives for competitive analysis.operations research and management science;

    Randomized algorithms for fully online multiprocessor scheduling with testing

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    We contribute the first randomized algorithm that is an integration of arbitrarily many deterministic algorithms for the fully online multiprocessor scheduling with testing problem. When there are two machines, we show that with two component algorithms its expected competitive ratio is already strictly smaller than the best proven deterministic competitive ratio lower bound. Such algorithmic results are rarely seen in the literature. Multiprocessor scheduling is one of the first combinatorial optimization problems that have received numerous studies. Recently, several research groups examined its testing variant, in which each job JjJ_j arrives with an upper bound uju_j on the processing time and a testing operation of length tjt_j; one can choose to execute JjJ_j for uju_j time, or to test JjJ_j for tjt_j time to obtain the exact processing time pjp_j followed by immediately executing the job for pjp_j time. Our target problem is the fully online version, in which the jobs arrive in sequence so that the testing decision needs to be made at the job arrival as well as the designated machine. We propose an expected (φ+3+1)(3.1490)(\sqrt{\varphi + 3} + 1) (\approx 3.1490)-competitive randomized algorithm as a non-uniform probability distribution over arbitrarily many deterministic algorithms, where φ=5+12\varphi = \frac {\sqrt{5} + 1}2 is the Golden ratio. When there are two machines, we show that our randomized algorithm based on two deterministic algorithms is already expected 3φ+3137φ4(2.1839)\frac {3 \varphi + 3 \sqrt{13 - 7\varphi}}4 (\approx 2.1839)-competitive. Besides, we use Yao's principle to prove lower bounds of 1.66821.6682 and 1.65221.6522 on the expected competitive ratio for any randomized algorithm at the presence of at least three machines and only two machines, respectively, and prove a lower bound of 2.21172.2117 on the competitive ratio for any deterministic algorithm when there are only two machines.Comment: 21 pages with 1 plot; an extended abstract to be submitte
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