Single Machine Precedence Constrained Scheduling Is a Vertex Cover Problem

In this paper we study the single machine precedence constrained scheduling problem of minimizing the sum of weighted completion time. Specifically, we settle an open problem first raised by Chudak and Hochbaum and whose answer was subsequently conjectured by Correa and Schulz. As shown by Correa and Schulz, the proof of this conjecture implies that the addressed scheduling problem is a special case of the vertex cover problem. This means that previous results for the scheduling problem can be explained, and in some cases improved, by means of vertex cover theory. For example, the conjecture implies the existence of a polynomial time algorithm for the special case of two-dimensional partial orders. This considerably extends Lawler’s result from 1978 for series-parallel orders.

[1]  W. Trotter,et al.  Combinatorics and Partially Ordered Sets: Dimension Theory , 1992 .

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

[3]  S. Safra,et al.  On the hardness of approximating minimum vertex cover , 2005 .

[4]  Ola Svensson,et al.  Approximating Precedence-Constrained Single Machine Scheduling by Coloring , 2006, APPROX-RANDOM.

[5]  Peter C. Fishburn,et al.  Partial orders of dimension 2 , 1972, Networks.

[6]  Ola Svensson,et al.  Scheduling with Precedence Constraints of Low Fractional Dimension , 2007, IPCO.

[7]  Gerhard J. Woeginger.  On the approximability of average completion time scheduling under precedence constraints , 2003, Discret. Appl. Math..

[8]  Vahab Mirrokni,et al.  Maximizing Non-Monotone Submodular Functions , 2007, FOCS 2007.

[9]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

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

[11]  Dorit S. Hochbaum,et al.  Efficient bounds for the stable set, vertex cover and set packing problems , 1983, Discret. Appl. Math..

[12]  Stavros G. Kolliopoulos,et al.  Partially-Ordered Knapsack and Applications to Scheduling , 2002, ESA.

[13]  O. Svensson,et al.  Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling , 2007, FOCS 2007.

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

[15]  Leslie E. Trotter,et al.  Properties of vertex packing and independence system polyhedra , 1974, Math. Program..

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

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

[18]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

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

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

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

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

[23]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..

[24]  George Steiner,et al.  Single Machine Scheduling with Precedence Constraints of Dimension 2 , 1984, Math. Oper. Res..

[25]  Andreas S. Schulz.  Scheduling to Minimize Total Weighted Completion Time: Performance Guarantees of LP-Based Heuristics and Lower Bounds , 1996, IPCO.

[26]  Nicolai N. Pisaruk,et al.  A fully combinatorial 2-approximation algorithm for precedence-constrained scheduling a single machine to minimize average weighted completion time , 2003, Discret. Appl. Math..

[27]  V. Paschos.  A survey of approximately optimal solutions to some covering and packing problems , 1997, CSUR.

[28]  Wayne E. Smith.  Various optimizers for single‐stage production , 1956 .

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

[30]  José R. Correa,et al.  Single-Machine Scheduling with Precedence Constraints , 2005, Math. Oper. Res..

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

[32]  David B. Shmoys,et al.  Scheduling to minimize average completion time: off-line and on-line algorithms , 1996, SODA '96.

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

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