The completion of this work has left a number of open questions which require further research. Suggestions for future work are discussed next.
Extension of the power of limited preemption from a single preemption to a …nite number of preemptions on uniform machines.
Investigation of the power of preemption for other classes of instances for the two unrelated machine problem, and an extension of that result to m number of machines.
Extension of the results on the power of splitting for the three uniform machine problem to other classes of instances, and a generalization of the bounds for m machines.
Study of the power of splitting for the problem of minimizing the sum of com-pletion times on m uniform machines.
Study of the power of splitting for the makespan and sum of completion times objectives on unrelated machines.
Arad, D., Mordechai, Y., & Shachnai, H. (2014). Tighter Bounds for Makespan Mini-mization on Unrelated Machines. arXiv preprint arXiv:1405.2530.
Atallah, M. J. (Ed.). (1998). Algorithms and theory of computation handbook. CRC press.
Bijsterbosch, J., & Volgenant, A. (2010). Solving the Rectangular assignment problem and applications. Annals of Operations Research, 181(1), 443-462.
Birkho¤, G. (1946). Three observations on linear algebra. Univ. Nac. Tucumán. Revista A, 5, 147-151.
Bourgeois, F., & Lassalle, J. C. (1971). An extension of the Munkres algorithm for the assignment problem to rectangular matrices. Communications of the ACM, 14(12), 802-804.
Braun, O., & Schmidt, G. (2003). Parallel Processor Scheduling with Limited Number of Preemptions. SIAM Journal on Computing, 32(3), 671-680.
Brucker, P. (2007). Scheduling Algorithms (3rd edition). Berlin: Springer.
Brucker, P., Hurink, J., Jurisch, B., & Wöstmann, B. (1997). A branch & bound algo-rithm for the open-shop problem. Discrete Applied Mathematics, 76(1), 43-59.
Burkard, R. E., & Cela, E. (1999). Linear assignment problems and extensions (pp.
75-149). Springer US.
Chen, B. (1991). Parametric bounds for LPT scheduling on uniform processors. Acta Mathematicae Applicatae Sinica, 7(1), 67-73.
Chen, B. (2004). Parallel scheduling for early completion. Handbook of scheduling:
algorithms, models, and performance analysis. CRC Press.
Chen, B., Potts, C. N., & Woeginger, G. J. (1999). A review of machine scheduling:
Complexity, algorithms and approximability. In Handbook of combinatorial optimiza-tion (pp. 1493-1641). Springer US.
Co¤man Jr, E. G., & Garey, M. R. (1993). Proof of the 4/3 Conjecture for Preemptive vs. Nonpreemptive two-processor scheduling. Journal of the ACM (JACM), 40(5), 991-1018.
Conway, R. W., Maxwell, W. L., & Miller, L. W. (1967). Theory of Scheduling, Palo Alto-London.
Cook, S. A. (1971, May). The complexity of theorem-proving procedures. In Proceedings of the third annual ACM symposium on Theory of computing (pp. 151-158). ACM.
Correa, J. R., Skutella, M., & Verschae, J. (2012). The Power of Preemption on Unre-lated Machines and Applications to Scheduling Orders. Mathematics of Operations Research, 37(2), 379-398.
Crescenzi, P., & Panconesi, A. (1991). Completeness in approximation classes. Infor-mation and Computation, 93(2), 241-262.
Dantzig, G. B., Orden, A., & Wolfe, P. (1955). The generalized simplex method for minimizing a linear form under linear inequality restraints. Paci…c Journal of Math-ematics, 5(2), 183-195.
Davis, E., & Ja¤e, J. M. (1981). Algorithms for scheduling tasks on unrelated proces-sors. Journal of the ACM (JACM), 28(4), 721-736.
De, P., & Morton, T. E. (1980). Scheduling to minimize makespan on unequal parallel processors. Decision Sciences, 11(4), 586-602.
Dobson, G. (1984). Scheduling independent tasks on uniform processors. SIAM Journal on Computing, 13(4), 705-716.
Edmonds, J., & Karp, R. M. (1972). Theoretical improvements in algorithmic e¢ ciency for network ‡ow problems. Journal of the ACM (JACM), 19(2), 248-264.
Epstein, L., Levin, A., Soper, A. J., Strusevich, V.A.(2016). Power of Preemption for Minimizing Total Completion Time on Uniform Machines.
Ford Jr, L. R., & Fulkerson, D. R. (1955). A simple algorithm for …nding maximal network ‡ows and an application to the Hitchcock problem (No. RAND/P-743).
RAND CORP SANTA MONICA CA.
Friesen, D. K. (1987). Tighter Bounds for LPT Scheduling on Uniform Processors.
SIAM Journal on Computing, 16(3), 554-560.
Gairing, M., Monien, B., & Woclaw, A. (2007). A faster combinatorial approximation algorithm for scheduling unrelated parallel machines. Theoretical Computer Science, 380(1), 87-99.
Garey, M. R., & Johnson, D. S. (1979). A Guide to the Theory of NP-Completeness.
San Francisco.
Gonzalez, T., & Sahni, S. (1976). Open shop scheduling to minimize …nish time. Journal of the ACM (JACM), 23(4), 665-679.
Gonzalez, T., Ibarra, O. H., & Sahni, S. (1977). Bounds for LPT schedules on uniform processors. SIAM Journal on Computing, 6(1), 155-166.
Gonzalez, T., & Sahni, S. (1978). Preemptive scheduling of uniform processor systems.
Journal of the ACM (JACM), 25(1), 92-101.
Gonzalez, T., Lawler, E. L., & Sahni, S. (1990). Optimal preemptive scheduling of two unrelated processors. ORSA Journal on Computing, 2(3), 219-224.
Graham, R. L. (1966). Bounds for Certain Multiprocessing Anomalies. Bell System Technical Journal, 45(9), 1563-1581.
Graham, R. L. (1969). Bounds on Multiprocessing Timing anomalies. SIAM Journal on Applied Mathematics, 17(2), 416-429.
Graham, R. L., Lawler, E. L., Lenstra, J. K., & Kan, A. H. G. (1979). Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of discrete Mathematics, 5, 287-326.
Hariri, A. M. A., & Potts, C. N. (1991). Heuristics for Scheduling Unrelated Parallel Machines. Computers & operations research, 18(3), 323-331.
Hochbaum, D. S., & Shmoys, D. B. (1987). Using dual approximation algorithms for scheduling problems theoretical and practical results. Journal of the ACM (JACM), 34(1), 144-162.
Horowitz, E., & Sahni, S. (1976). Exact and Approximate Algorithms for Scheduling Nonidentical Processors. Journal of the ACM (JACM), 23(2), 317-327.
Horvath, E. C., Lam, S., & Sethi, R. (1977). A Level Algorithm for Preemptive Schedul-ing. Journal of the ACM (JACM), 24(1), 32-43.
Ibarra, O. H., & Kim, C. E. (1977). Heuristic Algorithms for Scheduling Independent Tasks on Nonidentical Processors. Journal of the ACM (JACM), 24(2), 280-289.
Jiang, Y., Weng, Z., & Hu, J. (2014). Algorithms with Limited Number of Preemptions for Scheduling on Parallel Machines. Journal of Combinatorial Optimization, 27(4), 711-723.
Johnson, D. S., & Garey, M. R. (1979). Computers and intractability: A guide to the theory of NP-completeness. Freeman&Co, San Francisco, 32.
Karp, R. M. (1972). Reducibility Among Combinatorial Problems (pp. 85-103). Springer US.
Kellerer, H., Mansini, R., Pferschy, U., & Speranza, M. G. (2003). An e¢ cient fully polynomial approximation scheme for the subset-sum problem. Journal of Computer and System Sciences, 66(2), 349-370.
Khachiyan, L. G. (1980). Polynomial algorithms in linear programming. USSR Com-putational Mathematics and Mathematical Physics, 20(1), 53-72.
Kovács, A. (2006). Tighter Approximation Bounds for LPT Scheduling in Two Special Cases. In Algorithms and Complexity (pp. 187-198). Springer Berlin Heidelberg.
Kovács, A. (2010). New Approximation Bounds for LPT Scheduling. Algorithmica, 57(2), 413-433.
Kuhn, H. W. (1955). The Hungarian method for the assignment problem. Naval re-search logistics quarterly, 2(1-2), 83-97.
Lawler, E. L., & Labetoulle, J. (1978). On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming. Journal of the ACM (JACM), 25(4), 612-619.
Labetoulle, J., Lawler, E. L., Lenstra, J. K.,& Kan, A. R. (1982). Preemptive scheduling of uniform machines subject to release dates. In Progress in combinatorial optimiza-tion. Waterloo, Ont., 245-261
Lin, J. H., & Vitter, J. S. (1992, July). e-Approximations with minimum packing constraint violation. In Proceedings of the twenty-fourth annual ACM symposium on Theory of computing (pp. 771-782). ACM.
Lee, C. Y., & Strusevich, V. A. (2005). Two-machine Shop Scheduling with an Unca-pacitated Interstage Transporter. IIE Transactions, 37(8), 725-736.
Lenstra, J.K. & Rinnooy Kan, A.H.G. (1979), Computational Complexity of Discrete Optimization Problems, In: P.L. Hammer, E.L. Johnson and B.H. Korte, Editor(s), Annals of Discrete Mathematics, Elsevier, Volume 4, Pages 121-140, ISSN 0167-5060, ISBN 9780444853226
Lenstra, J. K., Shmoys, D. B., & Tardos, É. (1990). Approximation Algorithms for Scheduling Unrelated Parallel Machines. Mathematical programming, 46(1-3), 259-271.
Liu, J. W. S., & Liu, C. L. (1974). Performance analysis of heterogeneous multiprocessor computing systems. Computer architectures and networks (E. Gelenbe and R. Mahl, eds.), North Holland, 331-343.
Martello, S., Soumis, F., & Toth, P. (1997). Exact and approximation algorithms for makespan minimization on unrelated parallel machines. Discrete applied mathemat-ics, 75(2), 169-188.
McNaughton, R. (1959). Scheduling with deadlines and loss functions. Management Science, 6(1), 1-12.
Mokoto¤, E. (1999). Scheduling to minimize the makespan on identical parallel ma-chines: An LP-based algorithm. Investigacion Operative, 97-107.
Papadimitriou, C. H. (1981). On the Complexity of Integer Programming. Journal of the ACM (JACM), 28(4), 765-768.
Papadimitriou, C. H. (2003). Computational complexity. John Wiley and Sons Ltd..
Papadimitriou, C., & Yannakakis, M. (1988, January). Optimization, approximation, and complexity classes. In Proceedings of the twentieth annual ACM symposium on Theory of computing (pp. 229-234). ACM.
Pinedo, M. L. (2012). Scheduling: theory, algorithms, and systems. Springer.
Potts, C. N. (1985). Analysis of a linear programming heuristic for scheduling unrelated parallel machines. Discrete Applied Mathematics, 10(2), 155-164.
Potts, C. N., & Van Wassenhove, L. N. (1992). Integrating Scheduling with Batching and Lot-Sizing: A Review of Algorithms and Complexity. Journal of the Operational Research Society, 43(5), 395–406. http://doi.org/10.2307/2583559
Queyranne, M., & Sviridenko, M. (2002). A (2+")-approximation algorithm for the generalized preemptive open shop problem with minsum objective. Journal of Algo-rithms, 45(2), 202-212.
Rustogi, K., & Strusevich, V. A. (2013). Parallel Machine Scheduling: Impact of Adding Extra Machines. Operations Research, 61(5), 1243-1257.
Schulz, A. S., & Skutella, M. (2002). Scheduling Unrelated Machines by Randomized Rounding. SIAM J. Discret. Math., 15(4), 450–469.
http://doi.org/10.1137/S0895480199357078
Shmoys, D. B., & Tardos, É. (1993). An Approximation Algorithm for the Generalized Assignment Problem. Mathematical Programming, 62(1-3), 461-474.
Sera…ni, P. (1996). Scheduling Jobs on Several Machines with the Job Splitting prop-erty. Operations Research, 44(4), 617-628
Shor, N. Z. (1972). Utilization of the operation of space dilatation in the minimization of convex functions. Cybernetics and Systems Analysis, 6(1), 7-15.
Sitters, R. (2005). Complexity of preemptive minsum scheduling on unrelated parallel machines. Journal of Algorithms, 57(1), 37-48.
Sitters, R. A. (2008). Approximability of average completion time scheduling on unre-lated machines. In Algorithms-ESA 2008 (pp. 768-779). Springer Berlin Heidelberg.
Skutella, M. (2001). Convex quadratic and semide…nite programming relaxations in scheduling. Journal of the ACM, 48(2), 206–242.
http://doi.org/10.1145/375827.375840
Soper, A. J., & Strusevich, V. A. (2014a). Single parameter analysis of power of pre-emption on two and three uniform machines. Discrete Optimization, 12, 26-46.
Soper, A. J., & Strusevich, V. A. (2014b). Power of Preemption on Uniform Parallel Machines. Approximation, Randomization, and Combinatorial Optimization. Algo-rithms and Techniques (APPROX/RANDOM 2014)}, 28, 392-402.
Von Neumann, J. (1953). A certain zero-sum two-person game equivalent to the optimal assignment problem. Contributions to the Theory of Games, 2, 5-12.
Williamson, D. P., Hall, L. A., Hoogeveen, J. A., Hurkens, C. A. J., Lenstra, J. K., Sevast’Janov, S. V., & Shmoys, D. B. (1997). Short shop schedules. Operations Re-search, 45(2), 288-294.
Woeginger, G. J. (2000). A comment on scheduling on uniform machines under chain-type precedence constraints. Operations Research Letters, 26(3), 107-109.
Xing, W., & Zhang, J. (2000). Parallel machine scheduling with splitting jobs. Discrete Applied Mathematics, 103(1), 259-269.