Approximate Deadline-Scheduling with Precedence Constraints

We consider the classic problem of scheduling a set of n jobs non-preemptively on a singlemachine. Each job j has non-negative processing time, weight, and deadline, and a feasibleschedule needs to be consistent with chain-like precedence constraints. The goal is to computea feasible schedule that minimizes the sum of penalties of late jobs. Lenstra and Rinnoy Kan[Annals of Disc. Math., 1977] in their seminal work introduced this problem and showedthat it is strongly NP-hard, even when all processing times and weights are 1. We study theapproximability of the problem and our main result is an O(log k)-approximation algorithmfor instances with k distinct job deadlines.We also point out a surprising connection to a model for technology diffusion processes innetworks that was recently proposed by Goldberg and Liu [SODA, 2013]. In an instance ofsuch a problem one is given an undirected graph and a non-negative, integer threshold θ(v)for each of its vertices v. Vertices v in the graph are either active or inactive, and an inactivevertex v activates whenever it lies in component of size at least θ(v) in the graph induced byitself and all active vertices. The goal is now to find a smallest cardinality seed set of activevertices that leads to the activation of the entire graph.Goldberg and Liu showed that this problem has no o(log(n))-approximation algorithmsunless NP has quasi-polynomial time algorithms, and the authors presented an O(rk log(n))-approximation algorithm, where r is the radius of the given network, and k is the number ofdistinct vertex thresholds. The open question is whether the dependence of the approximationguarantee on r and k is avoidable. We answer this question affirmatively for instances where theunderlying graph is a spider. In such instances technology diffusion and precedence constrainedscheduling problem with unit processing times and weights are equivalent problems.