There has been recent success in using polyhedral formulations of scheduling problems not only to obtain good lower bounds in practice but also to develop provably good approximation algorithms. Most of these formulations rely on binary decision variables that are a kind of assignment variables. We present quite simple polynomialtime approximation algorithms that are based on linear programming formulations with completion time variables and give the best known performance guarantees for minimizing the total weighted completion time in several scheduling environments. This amplifies the importance of (appropriate) polyhedral formulations in the design of approximation algorithms with good worst-case performance guarantees.
In particular, for the problem of minimizing the total weighted completion time on a single machine subject to precedence constraints we present a polynomial-time approximation algorithm with performance ratio better than 2. This outperforms a (4 + ε)-approximation algorithm very recently proposed by Hall, Shmoys, and Wein that is based on time-indexed formulations. A slightly extended formulation leads to a performance guarantee of 3 for the same problem but with release dates. This improves a factor of 5.83 for the same problem and even the 4-approximation algorithm for the problem with release dates but without precedence constraints, both also due to Hall, Shmoys, and Wein.
By introducing new linear inequalities, we also show how to extend our technique to parallel machine problems. This leads, for instance, to the best known approximation algorithm for scheduling jobs with release dates on identical parallel machines. Finally, for the flow shop problem to minimize the total weighted completion time with both precedence constraints and release dates we present the first approximation algorithm that achieves a worst-case performance guarantee that is linear in the number of machines. We even extend this to multiprocessor flow shop scheduling.
The proofs of these results also imply guarantees for the lower bounds obtained by solving the proposed linear programming relaxations. This emphasizes the strength of linear programming formulations using completion time variables.