optimal job scheduling with prerequisites (DAG) and finish time penalties

Revision en5, by pabloskimg, 2019-07-22 08:07:52

There are $N$ jobs, each job $i$ has a single prerequisite job $P_i$ that must be done before, except for a global root job which has no prerequisite. Each job takes $T_i$ time to be finished, and if a job is finished at time $t$ it contributes with a penalty of $t * U_i$, where $U_i$ is the i-th job's penalty coefficient. What is the minimum penalty for finishing all jobs?

Constraints: $N <= 2 * 10^5$, everything is integer and non-negative

Notes:

• only a single task can be performed at a time (there is no concurrent work / task parallelism)
• the tasks DAG looks like a rooted tree

Any ideas on how to solve this? I was thinking of performing some kind of backtracking over all possible topological orderings of the DAG + some kind of extremely heavy pruning, but I haven't figured out yet a good pruning strategy to avoid exponential time. I also thought of using DP, but then I need to use bitmasks to keep track of unvisited nodes, which leads to exponential time.

#### History

Revisions

Rev. Lang. By When Δ Comment
en5 pabloskimg 2019-07-22 08:07:52 27
en4 pabloskimg 2019-07-22 08:05:53 47
en3 pabloskimg 2019-07-22 08:03:03 6 Tiny change: 'ng, but I still haven't f' -> 'ng, but I haven't f'
en2 pabloskimg 2019-07-22 08:02:00 510
en1 pabloskimg 2019-07-21 21:58:30 521 Initial revision (published)