On Structural Parameterizations of Graph Motif and Chromatic Number

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10156)

Abstract

Structural parameterizations for hard problems have proven to be a promising venture for discovering scenarios where the problem is tractable. In particular, when a problem is already known to be polynomially solvable for some class of inputs, then it is natural to parameterize by the distance of a general instance to a tractable class. In the context of graph algorithms, parameters like vertex cover, twin cover, treewidth and treedepth modulators, and distances to various special graph classes are increasingly popular choices for hard problems.

Our main focus in this work is the Graph Motif problem, which involves finding a connected induced subgraph in a vertex colored graph that respects a given palette. This problem is known to be hard in the traditional setting even for fairly structured classes of graphs. In particular, Graph Motif is known to be $$\mathsf {W}$$-hard when parameterized by the distance to split graphs, and para-$$\mathsf {NP}$$-hard when parameterized by the distance to co-graphs (which are the class of -free graphs). On the other hand, it is known to be $$\mathsf {FPT}$$ when parameterized by the distance to a clique or the distance to an independent set (or equivalently, vertex cover). Towards finding the boundary of tractability, we consider parameterizing the problem by the distance to threshold graphs, which are graphs that are both split and -free. Note that this is a natural choice of an intermediate parameter in that it is larger than the parameters for which the problem is hard and smaller than the ones for which the problem is tractable. Our main contribution is an $$\mathsf {FPT}$$ algorithm for the Graph Motif problem when parameterized by the distance to threshold graphs. We also address some related structural parameterizations for Chromatic Number. Here, we show that the problem admits a polynomial kernel when parameterized by the vertex deletion distance to a clique.

References

1. 1.
Ambalath, A.M., Balasundaram, R., Rao H., C., Koppula, V., Misra, N., Philip, G., Ramanujan, M.S.: On the kernelization complexity of colorful motifs. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 14–25. Springer, Heidelberg (2010). doi:
2. 2.
Betzler, N., Van Bevern, R., Fellows, M.R., Komusiewicz, C., Niedermeier, R.: Parameterized algorithmics for finding connected motifs in biological networks. IEEE/ACM TCBB 8(5), 1296–1308 (2011)Google Scholar
3. 3.
Bodlaender, H.L., Jansen, B.M., Kratsch, S.: Cross-composition: a new technique for kernelization lower bounds (2010). arXiv preprint arXiv:1011.4224
4. 4.
Bonnet, É., Sikora, F.: The graph motif problem parameterized by the structure of the input graph. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 43. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015)Google Scholar
5. 5.
Cai, L.: Parameterized complexity of vertex colouring. Discret. Appl. Math. 127(3), 415–429 (2003)
6. 6.
Chvátal, V., Hammer, P.L.: Aggregation of inequalities in integer programming. Ann. Discret. Math. 1, 145–162 (1977)
7. 7.
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)
8. 8.
Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms, vol. 4. Springer, New York (2015)
9. 9.
Downey, R.G., Fellows, M.R.: Parameterized Complexity, vol. 3. Springer, Heidelberg (1999)
10. 10.
Fellows, M.R., Fertin, G., Hermelin, D., Vialette, S.: Sharp tractability borderlines for finding connected motifs in vertex-colored graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 340–351. Springer, Heidelberg (2007). doi:
11. 11.
Foldes, S., Hammer, P.L.: Split graphs. Institut für Ökonometrie und Operations Research, Universität Bonn (1976)Google Scholar
12. 12.
Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Algorithmic lower bounds for problems parameterized by clique-width. In: Proceedings of 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 493–502. SIAM (2010)Google Scholar
13. 13.
Gallai, T.: Transitiv orientierbare graphen. Acta Mathematica Hungarica 18(1), 25–66 (1967)
14. 14.
Ganian, R.: Using neighborhood diversity to solve hard problems (2012). arXiv preprint arXiv:1201.3091
15. 15.
Ganian, R.: Improving vertex cover as a graph parameter. Discret. Math. Theoret. Comput. Sci. 17(2), 77–100 (2015)
16. 16.
Jansen, B.M., et al.: The power of data reduction: kernels for fundamental graph problems (2013)Google Scholar
17. 17.
Lacroix, V., Fernandes, C.G., Sagot, M.F.: Motif search in graphs: application to metabolic networks. IEEE/ACM TCBB 3(4), 360–368 (2006)Google Scholar
18. 18.
Mahadev, N.V., Peled, U.N.: Threshold Graphs and Related Topics, vol. 56. Elsevier, Amsterdam (1995)
19. 19.
Sæther, S.H., Telle, J.A.: Between treewidth and clique-width. Algorithmica 75(1), 218–253 (2016)
20. 20.
Tinhofer, G.: Bin-packing and matchings in threshold graphs. Discret. Appl. Math. 62(1), 279–289 (1995)

© Springer International Publishing AG 2017

Authors and Affiliations

• Bireswar Das
• 1
• Murali Krishna Enduri
• 1
Email author
• Neeldhara Misra
• 1
• I. Vinod Reddy
• 1
1. 1.IIT GandhinagarGandhinagarIndia

Personalised recommendations

Citepaper 