8,928 research outputs found

    Lift-and-Round to Improve Weighted Completion Time on Unrelated Machines

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    We consider the problem of scheduling jobs on unrelated machines so as to minimize the sum of weighted completion times. Our main result is a (3/2āˆ’c)(3/2-c)-approximation algorithm for some fixed c>0c>0, improving upon the long-standing bound of 3/2 (independently due to Skutella, Journal of the ACM, 2001, and Sethuraman & Squillante, SODA, 1999). To do this, we first introduce a new lift-and-project based SDP relaxation for the problem. This is necessary as the previous convex programming relaxations have an integrality gap of 3/23/2. Second, we give a new general bipartite-rounding procedure that produces an assignment with certain strong negative correlation properties.Comment: 21 pages, 4 figure

    Simultaneously Structured Models with Application to Sparse and Low-rank Matrices

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    The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and low-rank. Often norms that promote each individual structure are known, and allow for recovery using an order-wise optimal number of measurements (e.g., ā„“1\ell_1 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multi-objective optimization with these norms, then we can do no better, order-wise, than an algorithm that exploits only one of the present structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e. not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and low-rank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the ā„“1\ell_1 and nuclear norms requires many more measurements. This proves an order-wise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and low-rank tensor completion.Comment: 38 pages, 9 figure

    Beam search heuristics for the single machine scheduling problem with linear earliness and quadratic tardiness costs

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    In this paper, we consider the single machine scheduling problem with linear earliness and quadratic tardiness costs, and no machine idle time. We present heuristic algorithms based on the beam search technique. These algorithms include classic beam search procedures, as well as the filtered and recovering variants. Several dispatching rules are considered as evaluation functions, in order to analyse the effect of different rules on the effectiveness of the beam search algorithms. The computational results show that using better rules indeed improves the performance of the beam search heuristics. The detailed, filtered and recovering beam search procedures outperform the best existing heuristic. The best results are given by the recovering and detailed variants, which provide objective function values that are quite close to the optimum. For small to medium size instances, either of these procedures can be used. For larger instances, however, the detailed beam search algorithm requires excessive computation times, and the recovering beam search procedure then becomes the heuristic of choice.scheduling, single machine, linear earliness, quadratic tardiness, beam search, heuristics

    Single machine scheduling with exponential time-dependent learning effect and past-sequence-dependent setup times

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    AbstractIn this paper we consider the single machine scheduling problem with exponential time-dependent learning effect and past-sequence-dependent (p-s-d) setup times. By the exponential time-dependent learning effect, we mean that the processing time of a job is defined by an exponent function of the total normal processing time of the already processed jobs. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the quadratic job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions

    Better Unrelated Machine Scheduling for Weighted Completion Time via Random Offsets from Non-Uniform Distributions

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    In this paper we consider the classic scheduling problem of minimizing total weighted completion time on unrelated machines when jobs have release times, i.e, Rāˆ£rijāˆ£āˆ‘jwjCjR | r_{ij} | \sum_j w_j C_j using the three-field notation. For this problem, a 2-approximation is known based on a novel convex programming (J. ACM 2001 by Skutella). It has been a long standing open problem if one can improve upon this 2-approximation (Open Problem 8 in J. of Sched. 1999 by Schuurman and Woeginger). We answer this question in the affirmative by giving a 1.8786-approximation. We achieve this via a surprisingly simple linear programming, but a novel rounding algorithm and analysis. A key ingredient of our algorithm is the use of random offsets sampled from non-uniform distributions. We also consider the preemptive version of the problem, i.e, Rāˆ£rij,pmtnāˆ£āˆ‘jwjCjR | r_{ij},pmtn | \sum_j w_j C_j. We again use the idea of sampling offsets from non-uniform distributions to give the first better than 2-approximation for this problem. This improvement also requires use of a configuration LP with variables for each job's complete schedules along with more careful analysis. For both non-preemptive and preemptive versions, we break the approximation barrier of 2 for the first time.Comment: 24 pages. To apper in FOCS 201

    Fast approximation schemes for Boolean programming and scheduling problems related to positive convex Half-Product

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    We address a version of the Half-Product Problem and its restricted variant with a linear knapsack constraint. For these minimization problems of Boolean programming, we focus on the development of fully polynomial-time approximation schemes with running times that depend quadratically on the number of variables. Applications to various single machine scheduling problems are reported: minimizing the total weighted flow time with controllable processing times, minimizing the makespan with controllable release dates, minimizing the total weighted flow time for two models of scheduling with rejection
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