20,462 research outputs found

    Special cases of online parallel job scheduling

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    In this paper we consider the online scheduling of jobs, which require processing on a number of machines simultaneously. These jobs are presented to a decision maker one by one, where the next job becomes known as soon as the current job is scheduled. The objective is to minimize the makespan. For the problem with three machines we give a 2.8-competitive algorithm, improving upon the 3-competitive greedy algorithm. For the special case with arbitrary number of machines, where the jobs appear in non-increasing order of machine requirement, we give a 2.4815-competitive algorithm, improving the 2.75-competitive greedy algorithm

    Online Makespan Minimization with Parallel Schedules

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    In online makespan minimization a sequence of jobs σ=J1,...,Jn\sigma = J_1,..., J_n has to be scheduled on mm identical parallel machines so as to minimize the maximum completion time of any job. We investigate the problem with an essentially new model of resource augmentation. Here, an online algorithm is allowed to build several schedules in parallel while processing σ\sigma. At the end of the scheduling process the best schedule is selected. This model can be viewed as providing an online algorithm with extra space, which is invested to maintain multiple solutions. The setting is of particular interest in parallel processing environments where each processor can maintain a single or a small set of solutions. We develop a (4/3+\eps)-competitive algorithm, for any 0<\eps\leq 1, that uses a number of 1/\eps^{O(\log (1/\eps))} schedules. We also give a (1+\eps)-competitive algorithm, for any 0<\eps\leq 1, that builds a polynomial number of (m/\eps)^{O(\log (1/\eps) / \eps)} schedules. This value depends on mm but is independent of the input σ\sigma. The performance guarantees are nearly best possible. We show that any algorithm that achieves a competitiveness smaller than 4/3 must construct Ω(m)\Omega(m) schedules. Our algorithms make use of novel guessing schemes that (1) predict the optimum makespan of a job sequence σ\sigma to within a factor of 1+\eps and (2) guess the job processing times and their frequencies in σ\sigma. In (2) we have to sparsify the universe of all guesses so as to reduce the number of schedules to a constant. The competitive ratios achieved using parallel schedules are considerably smaller than those in the standard problem without resource augmentation

    Faster Algorithms for Semi-Matching Problems

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    We consider the problem of finding \textit{semi-matching} in bipartite graphs which is also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted case. For the weighted case, we give an O(nmlogn)O(nm\log n)-time algorithm, where nn is the number of vertices and mm is the number of edges, by exploiting the geometric structure of the problem. This improves the classical O(n3)O(n^3) algorithms by Horn [Operations Research 1973] and Bruno, Coffman and Sethi [Communications of the ACM 1974]. For the unweighted case, the bound could be improved even further. We give a simple divide-and-conquer algorithm which runs in O(nmlogn)O(\sqrt{n}m\log n) time, improving two previous O(nm)O(nm)-time algorithms by Abraham [MSc thesis, University of Glasgow 2003] and Harvey, Ladner, Lov\'asz and Tamir [WADS 2003 and Journal of Algorithms 2006]. We also extend this algorithm to solve the \textit{Balance Edge Cover} problem in O(nmlogn)O(\sqrt{n}m\log n) time, improving the previous O(nm)O(nm)-time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC 2008].Comment: ICALP 201

    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
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