236605 Advanced Topics Spring Semester, 2000

We shall denote the partial order (precedence relation) by , where i j means job i must be completed before job j starts. We assume the precedence relation is speciied in the form of a directed acyclic graph G = (V; E) (V is the set of nodes), in which there is a path from i to j ii i j. Goal: Find a non-preemptive schedule on a single machine which minimizes j w j C j. Our problem is NP-hard if we permit arbitrary precedence graphs. It remains NP-hard even if we additionally constrain processing times to be equal or weights to be equal (of course adding both constraints make the problem trivial). We use the following notation for any subset of jobs S N: p(S) = j2S p j p 2 (S) = j2S p 2 j w(S) = j2S w j w 2 (S) = j2S w 2 j 2 LP Formulations We present here several formulations of our problem.