Abstract
Motivated by applications in batch scheduling of jobs in manufacturing systems and distributed computing, we study two related problems. Given is a set of jobs {J 1,…,J n }, where J j has a processing time p j , and an undirected intersection graph G=({1,…,n},E), with an edge (i,j) whenever the pair of jobs J i and J j cannot be processed in the same batch. We are to schedule the jobs in batches, where each batch completes its processing when the last job in the batch completes execution. The goal is to minimize the sum of job completion times. Our two problems differ in the definition of completion time of a job within a given batch. In the first variant, a job completes its execution when its batch is completed, whereas in the second variant, a job completes execution when its own processing is completed.
For the first variant, we show that an adaptation of the greedy set cover algorithm gives a 4-approximation for perfect graphs. For the second variant, we give new or improved approximations for a number of different classes of graphs. The algorithms are of widely different genres (LP, greedy, subgraph covering), yet they curiously share a common feature in their use of randomized geometric partitioning.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bar-Noy, A., Bellare, M., Halldórsson, M.M., Shachnai, H., Tamir, T.: On chromatic sums and distributed resource allocation. Inf. Comput. 140(2), 183–202 (1998)
Bar-Noy, A., Halldórsson, M.M., Kortsarz, G., Salman, R., Shachnai, H.: Sum multicoloring of graphs. J. Algorithms 37(2), 422–450 (2000)
Brucker, P.: Scheduling Algorithms, 4th edn. Springer, Berlin (2004)
Chrobak, M., Kenyon-Mathieu, C.: Online algorithm column 10: competitiveness via doubling. SIGACT News. Dec. 2006
Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4, 233–235 (1979)
Corneil, D.G., Olariu, S., Stewart, L.: The ultimate interval graph recognition algorithm. In: Proceedings of the 9th ACM/SIAM Symposium on Discrete Algorithm (SODA98), pp. 175–180
Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)
Feige, U., Lovász, L., Tetali, P.: Approximating min sum set cover. Algorithmica 40(4), 219–234 (2004)
Frank, A.: On chain and antichain families of a partially ordered set. J. Comb. Theory Ser. B 29, 176–184 (1980)
Gandhi, R., Halldórsson, M.M., Kortsarz, G., Shachnai, H.: Improved bounds for sum multicoloring and scheduling dependent jobs with minsum criteria. In: 2nd Workshop on Approximation and Online Algorithms (WAOA2004), pp. 68–92
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. In: Topics on Perfect Graphs, North-Holland Math. Stud., vol. 88, pp. 325–356. North-Holland, Amsterdam (1984)
Halldórsson, M.M., Kortsarz, G., Shachnai, H.: Sum coloring interval and k-claw free graphs with application to scheduling dependent jobs. Algorithmica 37(3), 187–209 (2003)
Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM J. Discrete Math. 2, 68–72 (1989)
Ibarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22(4), 463–468 (1975)
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)
Munagala, K., Babu, S., Motwani, R., Widom, J.: The pipelined set cover problem. In: Proceedings of the 10th International Conference on Database Theory (ICDT2005), pp. 83–98
Nicoloso, S., Sarrafzadeh, M., Song, X.: On the sum coloring problem on interval graphs. Algorithmica 23(2), 109–126 (1999)
Sanders, P., Steurer, D.: An asymptotic approximation scheme for multigraph edge coloring. In: Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA2005), pp. 897–906
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics Series, vol. 24. Springer, New York (2003)
Smith, W.E.: Various optimizers for single-stage production. Nav. Res. Logist. Q. 3, 59–66 (1956)
Tanenbaum, A.S.: Distributed Operating Systems. Prentice-Hall, Englewood Cliffs (1995)
Woeginger, G.J.: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? INFORMS J. Comput. 12(1), 57–74 (2000)
Yannakakis, M., Gavril, F.: The maximum k-colorable subgraph problem for chordal graphs. Inf. Process. Lett. 24(2), 133–137 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
An extended abstract of this paper appeared in Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX ’06), pp. 116–127.
Rights and permissions
About this article
Cite this article
Epstein, L., Halldórsson, M.M., Levin, A. et al. Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs. Algorithmica 55, 643–665 (2009). https://doi.org/10.1007/s00453-007-9161-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-007-9161-z