We consider the online problem of scheduling jobs on identical machines, where jobs have precedence constraints. We are interested in the demanding setting where the jobs sizes are not known up-front, but are revealed only upon completion (the non-clairvoyant setting). Such precedence-constrained scheduling problems routinely arise in map-reduce and large-scale optimization. For minimizing the total weighted completion time, we give a constant-competitive algorithm. And for total weighted flow-time, we give an O(1/epsilon^2)-competitive algorithm under (1+epsilon)-speed augmentation and a natural "no-surprises" assumption on release dates of jobs (which we show is necessary in this context).
Our algorithm proceeds by assigning virtual rates to all waiting jobs, including the ones which are dependent on other uncompleted jobs. We then use these virtual rates to decide on the actual rates of minimal jobs (i.e., jobs which do not have dependencies and hence are eligible to run). Interestingly, the virtual rates are obtained by allocating time in a fair manner, using a Eisenberg-Gale-type convex program (which we can solve optimally using a primal-dual scheme). The optimality condition of this convex program allows us to show dual-fitting proofs more easily, without having to guess and hand-craft the duals. This idea of using fair virtual rates may have broader applicability in scheduling problems
