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This problem reminds me of what I have learnt in greedy algorithms, minimizing the entire completion time is a NP-complete problem. It's another way of phrasing bin-packing problem, which is used in memory allocation strategy.


That's a good point, but I bet the cases where it's hard don't come up in practice, so that it's possible to write something that performs near-optimal. Can anyone think of a realistic situation where not only is it hard to come up with a good balance of work; it's hard to even come up with a good approximation?


I am curious if there exists a polynomial time algorithm for finding within 2x of optimal for completion time, similar to how we could find within 2x of optimal for TSP using MST in 451. Or if anyone actually knows a reduction from minimizing completion time to TSP.

@mchoquet I believe that if you divide any workload up into small enough tasks, then you'll have a high probability of meeting your optimal work balance or at least within some reasonable approximation. You would need relatively few inherently sequential tasks that you have no way of predicting the approximate workload. I don't think this would be a very common problem, since there are not many tasks that are completely inherently sequential and large enough to cause workbalance complications and impossible to predict their approximate run time.


I think a more useful area to look if you wanted to convert minimizing completion time to a well-studied graph problem would be graph coloring algorithms, since scheduling is very similar to coloring a graph (colors are the processors, jobs are the nodes, and whatever restrictions are placed on the jobs are the edges)


I think here we are mostly just thinking of independent tasks, so the scheduling problem related to graph coloring is a little different (where edges indicate dependencies between jobs). It does, however, bring up the point of how much more complex balancing might be if we also had these restrictions. I know that finding even a constant factor approximation for graph coloring is NP-hard - not sure about bin-packing though.