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Preprint 683-2000

Combinatorial Optimization & Graph Algorithms group (COGA-Preprints)

Convex quadratic and semidefinite programming relaxations in Scheduling
Journal of the Association for Computing Machinery 48(2), 206-242, 2001.
primary: 90C27 Combinatorial optimization
secondary: 68Q25 Analysis of algorithms and problem complexity
90B35 Scheduling theory, deterministic
68M20 Performance evaluation; queueing; scheduling
90C59 Approximation methods and heuristics
90C20 Quadratic programming
90C25 Convex programming
Approximation algorithms, convex optimization, performance guarantee, randomized algorithms, scheduling theory, unrelated machines, worst-case ratio
We consider the problem of scheduling unrelated parallel machines subject to release dates so as to minimize the total weighted completion time of jobs. The main contribution of this paper is a provably good convex quadratic programming relaxation of strongly polynomial size for this problem. The best previously known approximation algorithms are based on LP relaxations in time- or interval-indexed variables. Those LP relaxations, however, suffer from a huge number of variables. As a result of the convex quadratic programming approach we can give a very simple and easy to analyze 2-approximation algorithm which can be further improved to performance guarantee 3/2 in the absence of release dates. We also consider preemptive scheduling problems and derive approximation algorithms and results on the power of preemption which improve upon the best previously known results for these settings. Finally, for the special case of two machines we introduce a more sophisticated semidefinite programming relaxation and apply the random hyperplane technique introduced by Goemans and Williamson for the MaxCut problem; this leads to an improved 1.2752-approximation.
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