Interval scheduling and colorful independent sets

Abstract

Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer \(k\), find a set of at least \(k\) pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains \(\mathrm{NP}\)-hard but admits constant ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-union graphs and on subclasses like strip graphs. Our investigations significantly benefit from a new structural “compactness” parameter of interval graphs and novel problem formulations using vertex-colored interval graphs. Our main contributions are as follows:

  1. 1.

    We show a complexity dichotomy: restricted to graph classes closed under induced subgraphs and disjoint unions, Independent Set is polynomial-time solvable if both input interval graphs are cluster graphs, and is \(\mathrm{NP}\)-hard otherwise.

  2. 2.

    We chart the possibilities and limits of effective polynomial-time preprocessing (also known as kernelization).

  3. 3.

    We extend Halldórsson and Karlsson (2006)’s fixed-parameter algorithm for Independent Set on strip graphs parameterized by the structural parameter “maximum number of live jobs” to show that the problem (also known as Job Interval Selection) is fixed-parameter tractable with respect to the parameter \(k\) and generalize their algorithm from strip graphs to 2-union graphs. Preliminary experiments with random data indicate that Job Interval Selection with up to 15 jobs and \(5\cdot 10^5\) intervals can be solved optimally in less than 5 min.

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Notes

  1. 1.

    The Exponential Time Hypothesis basically states that there is no \(2^{o(n)}\)-time algorithm for \(n\)-variable 3-SAT (Impagliazzo et al. 2001; Lokshtanov et al. 2011).

  2. 2.

    http://fpt.akt.tu-berlin.de/cis/.

References

  1. Alon, N., Yuster, R., & Zwick, U. (1995). Color-coding. Journal of the ACM, 42(4), 844–856.

    Article  Google Scholar 

  2. Bafna, V., Narayanan, B. O., & Ravi, R. (1996). Nonoverlapping local alignments (weighted independent sets of axis-parallel rectangles). Discrete Applied Mathematics, 71(1–3), 41–53.

    Article  Google Scholar 

  3. Bar-Yehuda, R., Halldórsson, M. M., Naor, J., Shachnai, H., & Shapira, I. (2006). Scheduling split intervals. SIAM Journal on Computing, 36(1), 1–15.

    Article  Google Scholar 

  4. Bodlaender, H. L., Jansen, B. M. P., & Kratsch, S. (2014). Kernelization lower bounds by cross-composition. SIAM Journal on Discrete Mathematics, 28(1), 277–305.

    Article  Google Scholar 

  5. Brandstädt, A., Le, V. B., & Spinrad, J. P. (1999). Graph classes: A survey. Philadelphia: SIAM.

    Google Scholar 

  6. Chen, J., Lu, S., Sze, S. H., Zhang, F. (2007). Improved algorithms for path, matching, and packing problems. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 298–307). New York: SIAM.

  7. Chuzhoy, J., Ostrovsky, R., & Rabani, Y. (2006). Approximation algorithms for the job interval selection problem and related scheduling problems. Mathematics of Operations Research, 31(4), 730–738.

    Article  Google Scholar 

  8. Corneil, D. G., Olariu, S., & Stewart, L. (2009). The LBFS structure and recognition of interval graphs. SIAM Journal on Discrete Mathematics, 23(4), 1905–1953.

    Article  Google Scholar 

  9. Downey, R. G., & Fellows, M. R. (2013). Fundamentals of parameterized complexity. New York: Springer.

    Google Scholar 

  10. Fellows, M. R., Gaspers, S., & Rosamond, F. A. (2012). Parameterizing by the number of numbers. Theory of Computing Systems, 50(4), 675–693.

    Article  Google Scholar 

  11. Flum, J., & Grohe, M. (2006). Parameterized complexity theory. Berlin: Springer.

    Google Scholar 

  12. Fulkerson, D. R., & Gross, O. A. (1965). Incidence matrices and interval graphs. Pacific Journal of Mathematics, 15(3), 835–855.

    Article  Google Scholar 

  13. Garey, M. R., Johnson, D. S., & Stockmeyer, L. (1976). Some simplified NP-complete graph problems. Theoretical Computer Science, 1(3), 237–267.

    Article  Google Scholar 

  14. Guo, J., & Niedermeier, R. (2007). Invitation to data reduction and problem kernelization. SIGACT News, 38(1), 31–45.

    Article  Google Scholar 

  15. Gyárfas, A., & West, D. B. (1995). Multitrack interval graphs. Congressus Numerantium, 109, 109–116.

    Google Scholar 

  16. Halldórsson, M. M., Karlsson, R. K. (2006). Strip graphs: Recognition and scheduling. In: Proceedings of the 32nd International Workshop on Graph-Theoretic Concepts in Computer Science. Lecture Notes in Computer Science (Vol. 4271, pp. 137–146). Berlin: Springer.

  17. Höhn, W., König, F. G., Möhring, R. H., & Lübbecke, M. E. (2011). Integrated sequencing and scheduling in coil coating. Management Science, 57(4), 647–666.

    Article  Google Scholar 

  18. Impagliazzo, R., Paturi, R., & Zane, F. (2001). Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4), 512–530.

    Article  Google Scholar 

  19. Jiang, M. (2010). On the parameterized complexity of some optimization problems related to multiple-interval graphs. Theoretical Computer Science, 411, 4253–4262.

    Article  Google Scholar 

  20. Jiang, M. (2013). Recognizing d-interval graphs and d-track interval graphs. Algorithmica, 66(3), 541–563.

    Article  Google Scholar 

  21. Kolen, A. W., Lenstra, J. K., Papadimitriou, C. H., & Spieksma, F. C. R. (2007). Interval scheduling: A survey. Naval Research Logistics, 54(5), 530–543.

    Article  Google Scholar 

  22. Koutis, I., Williams, R. (2009). Limits and applications of group algebras for parameterized problems. In: Proceedings of the 36th International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science (Vol. 5555, pp. 653–664). New York: Springer.

  23. Kratsch, S. (2014). Recent developments in kernelization: A survey. Bulletin of the European Association for Theoretical Computer Science, 113, 58–97.

    Google Scholar 

  24. Kung, H. T., Luccio, F., & Preparata, F. P. (1975). On finding the maxima of a set of vectors. Journal of the ACM, 22(4), 469–476.

    Article  Google Scholar 

  25. Lokshtanov, D., Marx, D., & Saurabh, S. (2011). Lower bounds based on the exponential time hypothesis. Bulletin of the European Association for Theoretical Computer Science, 105, 41–72.

    Google Scholar 

  26. Marx, D. (2011). Fixed-parameter tractable scheduling problems. In: Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091).

  27. Mnich, M., Wiese, A. (2014). Scheduling and fixed-parameter tractability. In: Proceedings of the 17th Conference on Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science (Vol. 8494, pp. 381–392). Berlin: Springer.

  28. Möhring, R. H. (2011). Algorithm engineering and industrial applications. Information Technology, 53(6), 302–311.

    Article  Google Scholar 

  29. Nakajima, K., & Hakimi, S. L. (1982). Complexity results for scheduling tasks with discrete starting times. Journal of Algorithms, 3(4), 344–361.

    Article  Google Scholar 

  30. Niedermeier, R. (2006). Invitation to fixed-parameter algorithms. Oxford: Oxford University Press.

  31. Scheinerman, E. (1988). Random interval graphs. Combinatorica, 8(4), 357–371.

    Article  Google Scholar 

  32. Schrijver, A. (2003). Combinatorial optimization: Polyhedra and efficiency (Vol. A). Berlin: Springer.

    Google Scholar 

  33. Spieksma, F. C. R. (1999). On the approximability of an interval scheduling problem. Journal of Scheduling, 2(5), 215–227.

    Article  Google Scholar 

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Acknowledgments

We thank Michael Dom and Hannes Moser for discussions on coil coating, which initiated our investigations on 2-Union Independent Set , as well as Wiebke Höhn for providing details regarding the application of 2-Union Independent Set  in steel manufacturing. René van Bevern was supported by the Deutsche Forschungsgemeinschaft (DFG), project DAPA, NI 369/12. Part of the work was done while being supported by DFG project AREG, NI 369/9. Mathias Weller was supported by the DFG, project DARE, NI 369/11.

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Correspondence to René van Bevern.

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A preliminary version of this article appeared in the proceedings of the 23rd International Symposium on Algorithms and Computation (ISAAC 2012), volume 7676 in Lecture Notes in Computer Science, pp. 247–256, Springer, 2012. Besides providing full proof details, this revised and extended version improves running times, shows that Job Interval Selection is fixed-parameter tractable with respect to the standard parameter \(k\), and introduces the parameter \(c\)-compactness. Moreover, it adds an experimental evaluation of the algorithms.

Mathias Weller: Main work done while affiliated with Institut für Softwaretechnik und Theoretische Informatik, TU Berlin.

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van Bevern, R., Mnich, M., Niedermeier, R. et al. Interval scheduling and colorful independent sets. J Sched 18, 449–469 (2015). https://doi.org/10.1007/s10951-014-0398-5

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Keywords

  • Interval graphs
  • 2-union graphs
  • Strip graphs
  • Job interval selection
  • Parameterized complexity