On the approximability of average completion time scheduling under precedence constraints

We consider the scheduling problem of minimizing the average weighted job completion time on a single machine under precedence constraints. We show that this problem with arbitrary job weights, the special case of the problem where all job weights are one, and several other special cases of the problem all have the same approximability threshold with respect to polynomial time approximation algorithms. Moreover, for the special case of interval order precedence constraints and for the special case of convex bipartite precedence constraints, we give a polynomial time approximation algorithm with worst case performance guarantee arbitrarily close to the golden ratio 2/1 (1 + √5) ≅ 1:61803.

[1]  Martin E. Dyer,et al.  Formulating the single machine sequencing problem with release dates as a mixed integer program , 1990, Discret. Appl. Math..

[2]  Gerhard J. Woeginger,et al.  Polynomial time approximation algorithms for machine scheduling: ten open problems , 1999 .

[3]  E. Lawler.  Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints , 1978 .

[4]  C. N. Potts,et al.  An algorithm for the single machine sequencing problem with precedence constraints , 1980 .

[5]  Jeremy P. Spinrad,et al.  Bipartite permutation graphs , 1987, Discret. Appl. Math..

[6]  Fabián A. Chudak,et al.  A half-integral linear programming relaxation for scheduling precedence-constrained jobs on a single machine , 1999, Oper. Res. Lett..

[7]  Jan Karel Lenstra,et al.  Complexity of Scheduling under Precedence Constraints , 1978, Oper. Res..

[8]  Eugene L. Lawler,et al.  Sequencing and scheduling: algorithms and complexity , 1989 .

[9]  David P. Williamson,et al.  Two-dimensional Gantt charts and a scheduling algorithm of Lawler , 1999, SODA '99.

[10]  Rajeev Motwani,et al.  Precedence Constrained Scheduling to Minimize Sum of Weighted Completion Times on a Single Machine , 1999, Discret. Appl. Math..

[11]  Jeffrey B. Sidney,et al.  Decomposition Algorithms for Single-Machine Sequencing with Precedence Relations and Deferral Costs , 1975, Oper. Res..

[12]  Rolf H. Möhring,et al.  Computationally Tractable Classes of Ordered Sets , 1989 .

[13]  E.L. Lawler,et al.  Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey , 1977 .

[14]  David B. Shmoys,et al.  Scheduling to Minimize Average Completion Time: Off-Line and On-Line Approximation Algorithms , 1997, Math. Oper. Res..

[15]  Maurice Queyranne,et al.  Decompositions, Network Flows, and a Precedence Constrained Single-Machine Scheduling Problem , 2003, Oper. Res..

[16]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .