On capacitated set cover problems
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
A primal-dual approximation algorithm for min-sum single-machine scheduling problems
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Online scheduling with general cost functions
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
An FPTAS for the minimum total weighted tardiness problem with a fixed number of distinct due dates
ACM Transactions on Algorithms (TALG)
Weighted geometric set multi-cover via quasi-uniform sampling
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
The complexity of scheduling for p-norms of flow and stretch
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
Dual techniques for scheduling on a machine with varying speed
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
Exact algorithms and APX-hardness results for geometric packing and covering problems
Computational Geometry: Theory and Applications
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We consider the following general scheduling problem: The input consists of $n$ jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as weighted flow, weighted tardiness, and sum of flow squared. The main contribution of this paper is a randomized polynomial-time algorithm with an approximation ratio $O(\log \log nP )$, where $P$ is the maximum job size. We also give an $O(1)$ approximation in the special case when all jobs have identical release times. Initially, we show how to reduce this scheduling problem to a particular geometric set-cover problem. We then consider a natural linear programming formulation of this geometric set-cover problem, strengthened by adding knapsack cover inequalities, and show that rounding the solution of this linear program can be reduced to other particular geometric set-cover problems. We then develop algorithms for these sub-problems using the local ratio technique, and Varadarajan's quasi-uniform sampling technique. This general algorithmic approach improves the best known approximation ratios by at least an exponential factor (and much more in some cases) for essentially all of the nontrivial common special cases of this problem. We believe that this geometric interpretation of scheduling is of independent interest.