A simple parallel algorithm for the maximal independent set problem
SIAM Journal on Computing
The mathematics of nonlinear programming
The mathematics of nonlinear programming
Randomized rounding: a technique for provably good algorithms and algorithmic proofs
Combinatorica - Theory of Computing
Approximation algorithms for scheduling unrelated parallel machines
Mathematical Programming: Series A and B
Approximating linear programming is log-space complete for P
Information Processing Letters
Simulating (logcn)-wise independence in NC
Journal of the ACM (JACM)
A note on approximate linear programming
Information Processing Letters
A parallel approximation algorithm for positive linear programming
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
The probabilistic method yields deterministic parallel algorithms
Proceedings of the 30th IEEE symposium on Foundations of computer science
An approximation algorithm for the generalized assignment problem
Mathematical Programming: Series A and B
New ${\bf \frac{3}{4}}$-Approximation Algorithms for the Maximum Satisfiability Problem
SIAM Journal on Discrete Mathematics
Paradigms for fast parallel approximability
Paradigms for fast parallel approximability
Exact and Approximate Algorithms for Scheduling Nonidentical Processors
Journal of the ACM (JACM)
Approximability of flow shop scheduling
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Linear time approximation schemes for parallel processor scheduling
SPDP '96 Proceedings of the 8th IEEE Symposium on Parallel and Distributed Processing (SPDP '96)
Computers and Operations Research
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We show how to approximate in NC the problem of scheduling unrelated parallel machines, for a fixed number of machines in which the makespan Cmax is the objective function to minimize. We develop an approximate NC algorithm which finds a schedule whose length is at most \[(1+o(1))\big(C^{\ast}_{\max} + \sqrt{3\cdot C^{\ast}_{\max}\ln(2n(n-1)/\varepsilon)}\big),\] where C*max denotes the optimal schedule, n the total number of jobs and ε a small positive constant. Our approach shows how to relate the linear program obtained by relaxing the integer programming formulation of the problem with a linear program formulation that is positive and in the packing/covering form. The established relationship enables us to transfer approximate fractional solutions from the later formulation that is known to be approximable in NC. Then, we show how to obtain an integer approximate solution, i.e. a schedule, from the fractional one, using the randomized rounding technique. We stress that our analysis assumes that the length of the schedule is Ω(ln n) and that the min pij and max pij values are not too disparate (where pij is the time to run job j on machine i).Finally, we show that the same technique can be applied to the general assignment problem with a fixed number of machines and makespan T.