Or 625 — 3/4/2004 — Scribe: Tom Wexler

Recall from last time that we are working towards a 2-approximation algorithm for 1|prec|∑wjCj. We will focus on initial sets S, which have the property that if j ∈ S and k → j then k ∈ S. We defined ρ(S) = w(S) p(S) , and S ∗ to be an initial set with maximum ρ(S) over all initial sets S. Our algorithm finds S∗, finds a 2-approximate schedule for S∗ respecting precedence constraints, and then recurses on N−S∗, appending the recursively constructed solution. Last time we showed that any feasible schedule of S∗ was within a factor of 2 to the optimal, so now all we need to prove is the following result.