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Weighted interval scheduling with m-machines ('Weighted interval scheduling with m-machines')

I encountered the problem of weighted interval scheduling on m identical machines (as discussed in the link above). My greedy solution is exactly the same as the answer provided by D.W. in that link. 1. sort the interval by profit-to-duration in decreasing order, 2. then select intervals in that order. If there is still an available machine left, select it; Otherwise, skip the currently scanned interval. The selection is done when all intervals are considered.

I found the greedy solution provided pretty good empirical performance, and I wish to derive a theoretical approximation factor for this greedy solution. From the empirical result, I guess the approximation factor is very close to 1. I read about how to prove the approximation factor for greedy solutions in most general cases, but I can not transfer that knowledge to my case here. I think it is necessary to derive certain characteristics out of my greedy solution (in a mathematical form), which I can't :(.

In short, I would like to know how to derive an approximation factor for my greedy solution.

EddieG
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Unfortunately, the approximation guarantee of the algorithm is unbounded.

For example, Consider a single machine and only two intervals: interval $I_1$ with profit $1$ and length $1$; and interval $I_2$ with profit $w \gg 1$, and length $w+1$. Suppose the two intervals overlap. Then, the greedy algorithm gives a solution of profit $1$, i.e., interval $I_1$. Whereas, the optimal solution is of profit $w$, i.e., interval $I_2$.

Inuyasha Yagami
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