Minimum Reload Cost Graph Factors

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11376)

Abstract

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is the smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem to that of finding an r-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of $$r=2$$, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of r and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W-hardness results and present an FPT-algorithm.

Keywords

Parameterized complexity Graph factors Reload costs

References

1. 1.
Galbiati, G.: The complexity of a minimum reload cost diameter problem. Discrete Appl. Math. 156(18), 3494–3497 (2008)
2. 2.
Arkoulis, S., Anifantis, E., Karyotis, V., Papavassiliou, S., Mitrou, N.: On the optimal, fair and channel-aware cognitive radio network reconfiguration. Comput. Netw. 57(8), 1739–1757 (2013)
3. 3.
Gözüpek, D., Buhari, S., Alagöz, F.: A spectrum switching delay-aware scheduling algorithm for centralized cognitive radio networks. IEEE Trans. Mob. Comput. 12(7), 1270–1280 (2013)
4. 4.
Celik, A., Kamal, A.E.: Green cooperative spectrum sensing and scheduling in heterogeneous cognitive radio networks. IEEE Trans. Cogn. Commun. Netw. 2(3), 238–248 (2016)
5. 5.
Wirth, H.C., Steffan, J.: Reload cost problems: minimum diameter spanning tree. Discrete Appl. Math. 113(1), 73–85 (2001)
6. 6.
Gourvès, L., Lyra, A., Martinhon, C., Monnot, J.: The minimum reload s-t path, trail and walk problems. Discrete Appl. Math. 158(13), 1404–1417 (2010)
7. 7.
Amaldi, E., Galbiati, G., Maffioli, F.: On minimum reload cost paths, tours, and flows. Networks 57(3), 254–260 (2011)
8. 8.
Galbiati, G., Gualandi, S., Maffioli, F.: On minimum changeover cost arborescences. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 112–123. Springer, Heidelberg (2011).
9. 9.
Gözüpek, D., Shalom, M., Voloshin, A., Zaks, S.: On the complexity of constructing minimum changeover cost arborescences. Theor. Comput. Sci. 540, 40–52 (2014)
10. 10.
Gözüpek, D., Shachnai, H., Shalom, M., Zaks, S.: Constructing minimum changeover cost arborescenses in bounded treewidth graphs. Theor. Comput. Sci. 621, 22–36 (2016)
11. 11.
Gözüpek, D., Özkan, S., Paul, C., Sau, I., Shalom, M.: Parameterized complexity of the mincca problem on graphs of bounded decomposability. Theor. Comput. Sci. 690, 91–103 (2017)
12. 12.
Gözüpek, D., Shalom, M.: Edge coloring with minimum reload/changeover costs. arXiv preprint arXiv:1607.06751 (2016)
13. 13.
Gamvros, I., Gouveia, L., Raghavan, S.: Reload cost trees and network design. Networks 59(4), 365–379 (2012)
14. 14.
Meijer, H., Núñez-Rodríguez, Y., Rappaport, D.: An algorithm for computing simple k-factors. Inf. Process. Lett. 109(12), 620–625 (2009)
15. 15.
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 24. Springer, Heidelberg (2003)
16. 16.
Galbiati, G., Gualandi, S., Maffioli, F.: On minimum reload cost cycle cover. Discrete Appl. Math. 164, 112–120 (2014)
17. 17.
Baste, J., Gözüpek, D., Shalom, M., Thilikos, D.M.: Minimum reload cost graph factors. CoRR abs/1810.11700 (2018). http://arxiv.org/abs/1810.11700
18. 18.
Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015).
19. 19.
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. TCS. Springer, London (2013).
20. 20.
Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. Comput. Syst. Sci. 67(4), 757–771 (2003)
21. 21.
Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 150–159. IEEE Computer Society (2011)Google Scholar
22. 22.
Kloks, T. (ed.): Treewidth. LNCS, vol. 842. Springer, Heidelberg (1994).
23. 23.
Pulleyblank, W.R.: Faces of matching polyhedra. Ph.D. thesis, University of Waterloo (1973)Google Scholar

© Springer Nature Switzerland AG 2019

Authors and Affiliations

1. 1.Institut für Optimierung und Operations ResearchUniversität UlmUlmGermany
2. 2.Department of Computer EngineeringGebze Technical UniversityGebzeTurkey
3. 3.TelHai CollegeUpper GalileeIsrael
4. 4.AlGCo project-team, LIRMM, CNRS, Université de MontpellierMontpellierFrance
5. 5.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece

Personalised recommendations

Citepaper 