Chrobak Scheduling

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Idea: use LP duality to get lower bounds.

primal:

maximize ∑_jt x_jt w_j subject to (∀ t) ∑_{j∼ t} x_jt ≤ 1, (∀ j) ∑_{t ∼ j} x_jt ≤ 1

Here j indexes the jobs, t indexes the time slots. x_jt 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 y_j + ∑_t z_t subject to (∀ t,j : t∼ j) y_j + z_t ≥ w_j, y_j ≥ 0, z_t ≥ 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 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