Skip to main content
SpringerLink
Log in

Vertex Cover Meets Scheduling

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We consider a hybrid two-stage optimization problem that generalizes two classic combinatorial optimization problems: (i) weighted vertex cover in graphs, and (ii) makespan minimization in multiprocessor scheduling. An instance specifies a machine environment, a set of jobs, and an undirected graph over the jobs. The goal is to select a subset of the jobs that forms a vertex cover and to schedule it on a set of parallel machines so as to minimize the makespan. We call this problem family vertex cover meets multiprocessor scheduling (VCMS). The problem is motivated by networks where vertices represent servers and each edge corresponds to a task that can be done by any of the two servers of its endpoint. Each selected server can complete any number of tasks assigned to it within a given time defined by its weight, as the time consumption of the server is roughly equal to its activation time. The activation is performed by a set of \(m\) processors, such that every selected server is activated by one processor, and the goal is to minimize the maximum total activation time assigned to any processor. We design a multitude of approximation algorithms for VCMS and its variants, many of which match or almost match the best approximation bound known for the vertex cover problem. In particular, we give a \(2\)-approximation for the case of a fixed number of unrelated machines, a \(3\)-approximation for an arbitrary number of unrelated machines, and a \((2+\varepsilon )\)-approximation for an arbitrary number of identical machines. Furthermore we consider special graph classes for which the weighted vertex cover problem can be solved to optimality in polynomial time: for many of these classes, there is a PTAS for VCMS on identical machines; for bipartite graphs, however, VCMS on identical machines turns out to be APX-hard. Finally, we study the bin packing counterpart of VCMS and design a \((2+\varepsilon )\)-approximation algorithm for it.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. It is sufficient to count vertices of \(T\) whose labels have cardinality \(\omega +1\). Every such vertex of \(T\) has a vertex \(v \in V\) in its label that its parent does not have, and every vertex \(v \in V\) has at most one such vertex of \(T\).

References

  1. Azar, Y., Naor, J., Rom, R.: The competitiveness of on-line assignments. J. Algorithms 18(2), 221–237 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bar-Yehuda, R., Even, S.: A linear-time approximation algorithm for the weighted vertex cover problem. J. Algorithms 2(2), 198–203 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  3. Corneil, D.G., Rotics, U.: On the relationship between clique-width and treewidth. SIAM J. Comput. 34(4), 825–847 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162(1), 439–485 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Espelage, W., Gurski, F., Wanke, E.: Deciding clique-width for graphs of bounded tree-width. J. Graph Algorithms Appl. 7(2), 141–180 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fernandez de la Vega, W., Lueker, G.S.: Bin packing can be solved within \(1+\varepsilon \) in linear time. Combinatorica 1, 349–355 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Graham, R.L.: Bounds for certain multiprocessor anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)

    Article  Google Scholar 

  8. Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17, 416–429 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hochbaum, D.S.: Approximation algorithms for the set covering and vertex cover problems. SIAM J. Comput. 11(3), 555–556 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hochbaum, D.S., Megiddo, N., Naor, J., Tamir, A.: Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality. Math. Program. 62, 69–83 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems: theoretical and practical results. J. ACM 34(1), 144–162 (1987)

    Article  MathSciNet  Google Scholar 

  13. Hochbaum, D.S., Shmoys, D.B.: A polynomial approximation scheme for scheduling on uniform processors: using the dual approximation approach. SIAM J. Comput. 17(3), 539–551 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. J. ACM 23(2), 317–327 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., Graham, R.L.: Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM J. Computi. 3, 256–278 (1974)

    MathSciNet  Google Scholar 

  16. Karmarkar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional bin-packing problem. In: Proceedings of the 23rd annual symposium on foundations of computer science (FOCS’82), pp 312–320 (1982)

  17. Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci. 74(3), 335–349 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46, 259–271 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lenstra Jr, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nemhauser, G.L., Trotter Jr, L.E.: Vertex packings: structural properties and algorithms. Math. Program. 8(1), 232–248 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nip, K., Wang, Z.: Combination of two-machine flow shop scheduling and shortest path problems. In: Proc. of the 19th international conference on computing and combinatorics (COCOON’13), pp 680–687 (2013)

  22. Nip, K., Wang, Z., Nobibon, F.T., Leus, R.: A combination of flow shop scheduling and the shortest path problem. J. Comb. Optim. 29(1), 36–52 (2015)

    Article  MathSciNet  Google Scholar 

  23. Shmoys, D.B., Tardos, É.: An approximation algorithm for the generalized assignment problem. Math. Program. 62, 461–474 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, Z., Cui, Z.: Combination of parallel machine scheduling and vertex cover. Theor. Comput. Sci. 460, 10–15 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, Z., Hong, W., He, D.: A combination of parallel machine scheduling and the covering problem. Pac. J. Optim. 10, 577–591 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Haifa, Haifa, Israel

    Leah Epstein

  2. Faculty of IE&M, The Technion, Haifa, Israel

    Asaf Levin

  3. Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, The Netherlands

    Gerhard J. Woeginger

Authors
  1. Leah Epstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Asaf Levin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gerhard J. Woeginger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leah Epstein.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Epstein, L., Levin, A. & Woeginger, G.J. Vertex Cover Meets Scheduling. Algorithmica 74, 1148–1173 (2016). https://doi.org/10.1007/s00453-015-9992-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-015-9992-y

Keywords

Search

Navigation