Chrobak Scheduling

ClassS04CS141 | recent changes | Preferences

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Idea: use LP duality to get lower bounds.

primal:

maximize jt xjt wj subject to (∀ t) j∼ t xjt ≤ 1, (∀ j) t ∼ j xjt ≤ 1

Here j indexes the jobs, t indexes the time slots. xjt indicates whether job j is scheduled at time t. t ∼ j indicates that job j is allowed to be scheduled at time t.

dual:

minimize j yj + t zt subject to (∀ t,j : t∼ j) yj + zt ≥ wj, yj ≥ 0, zt ≥ 0

the cost of any feasible dual solution is an upper bound on opt.

Note: the integrality gap is 0, since this is a special case of maximum-weight matching.

greedy on-line is (1/2)-competitive

greedy:

 1. At each time step t:
 2.    Assign to time slot t the maximum-weight job j not yet assigned.
 3.    For the analysis, set yj = zj = wj.

thm: The above algorithm is 2-competitive
Proof:

The dual solution is feasible, and its cost is twice the cost of the matching found.
QED

ClassS04CS141 | recent changes | Preferences
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Edited February 2, 2004 6:20 pm by Neal (diff)
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