Abstract
We continue the investigation proposed in [COCOA 2016, Weller, Chateau, Giroudeau, König and Pollet “On Residual Approximation in Solution Extension Problems”] about the study of extended problems. In this context, a partial feasible solution is given in advance and the goal is to extend this partial solution. In this paper, we focus on the edge-weighted spanning star forest problem for both minimization and maximization versions. The goal here is to find a vertex partition of an edge-weighted complete graph into disjoint non-trivial stars and the value of a solution is given by the sum of the edge-weights of the stars. We propose NP-hardness, parameterized complexity, positive and negative approximation results.
K. Khoshkhah, D.O. Theis supported by the Estonian Research Council (PUT Exploratory Grant #620); M. Khosravian Ghadikolaei supported by a DORA Plus scholarship of the Archimedes Foundation (funded by the European Regional Development Fund).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In this case, it is also required that some vertices are forbidden.
References
Agatz, N.A.H., Erera, A.L., Savelsbergh, M.W.P., Wang, X.: Optimization for dynamic ride-sharing: a review. Eur. J. Oper. Res. 223(2), 295–303 (2012)
Agra, A., Cardoso, D., Cerfeira, O., Rocha, E.: A spanning star forest model for the diversity problem in automobile industry. In: ECCO XVIII, Minsk (2005)
Athanassopoulos, S., Caragiannis, I., Kaklamanis, C., Papaioannou, E.: Energy-efficient communication in multi-interface wireless networks. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 102–111. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03816-7_10
Böckenhauer, H.-J., Hromkovic, J., Klasing, R., Seibert, S., Unger, W.: Approximation algorithms for the TSP with sharpened triangle inequality. Inf. Process. Lett. 75(3), 133–138 (2000)
Chakrabarty, D., Goel, G.: On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP. SIAM J. Comput. 39(6), 2189–2211 (2010)
Chen, N., Engelberg, R., Nguyen, C.T., Raghavendra, P., Rudra, A., Singh, G.: Improved approximation algorithms for the spanning star forest problem. Algorithmica 65(3), 498–516 (2013)
Delbot, F., Laforest, C., Phan, R.: Graphs with forbidden and required vertices. In: ALGOTEL 2015-17emes Rencontres Francophones sur les Aspects Algorithmiques des Télécommunications Jun 2015, Beaune (2015). https://hal.archives-ouvertes.fr/hal-01148233
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London (2013). https://doi.org/10.1007/978-1-4471-5559-1
Fotakis, D., Gourvès, L., Monnot, J.: Conference program design with single-peaked and single-crossing preferences. In: Cai, Y., Vetta, A. (eds.) WINE 2016. LNCS, vol. 10123, pp. 221–235. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-54110-4_16
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Hartman, I.B.-A., Keren, D., Dbai, A.A., Cohen, E., Knapen, L., Yasar, A.-U.-H., Janssens, D.: Theory and practice in large carpooling problems. In: Shakshuki, E.M., Yasar, A.-U.-H. (eds.) Proceedings of the 5th International Conference on Ambient Systems, Networks and Technologies (ANT 2014), The 4th International Conference on Sustainable Energy Information Technology (SEIT-2014), Hasselt, 2–5 June 2014, vol. 32. Procedia Computer Science, pp. 339–347. Elsevier (2014)
He, J., Liang, H.: Improved approximation for spanning star forest in dense graphs. J. Comb. Optim. 25(2), 255–264 (2013)
Kutiel, G.: Approximation algorithms for the maximum carpool matching problem. In: Weil, P. (ed.) CSR 2017. LNCS, vol. 10304, pp. 206–216. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58747-9_19
Nguyen, C.T., Shen, J., Hou, M., Sheng, L., Miller, W., Zhang, L.: Approximating the spanning star forest problem and its application to genomic sequence alignment. SIAM J. Comput. 38(3), 946–962 (2008)
Nguyen, V.H.: The maximum weight spanning star forest problem on cactus graphs. Discrete Math. Algorithms Appl. 7(2), 1550018 (2015)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)
Tuza, Z.: Graph colorings with local constraints - a survey. Discussiones Mathematicae Graph Theory 17(2), 161–228 (1997)
Weller, M., Chateau, A., Giroudeau, R., König, J.-C., Pollet, V.: On residual approximation in solution extension problems. In: Chan, T.-H.H., Li, M., Wang, L. (eds.) COCOA 2016. LNCS, vol. 10043, pp. 463–476. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-48749-6_34
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Khoshkhah, K., Khosravian Ghadikolaei, M., Monnot, J., Theis, D.O. (2017). Extended Spanning Star Forest Problems. In: Gao, X., Du, H., Han, M. (eds) Combinatorial Optimization and Applications. COCOA 2017. Lecture Notes in Computer Science(), vol 10627. Springer, Cham. https://doi.org/10.1007/978-3-319-71150-8_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-71150-8_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-71149-2
Online ISBN: 978-3-319-71150-8
eBook Packages: Computer ScienceComputer Science (R0)