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The Balanced Connected Subgraph Problem

  • Sujoy Bhore
  • Sourav Chakraborty
  • Satyabrata Jana
  • Joseph S. B. Mitchell
  • Supantha PanditEmail author
  • Sasanka Roy
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)

Abstract

The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following Open image in new window (shortly, Open image in new window ) problem. The input is a graph \(G=(V,E)\), with each vertex in the set V having an assigned color, “ Open image in new window ” or “ Open image in new window ”. We seek a maximum-cardinality subset \(V'\subseteq V\) of vertices that is Open image in new window (having exactly \(|V'|/2\) red nodes and \(|V'|/2\) blue nodes), such that the subgraph induced by the vertex set \(V'\) in G is connected. We show that the BCS problem is NP-hard, even for bipartite graphs G (with red/blue color assignment not necessarily being a proper 2-coloring). Further, we consider this problem for various classes of the input graph G, including, e.g., planar graphs, chordal graphs, trees, split graphs, bipartite graphs with a proper red/blue 2-coloring, and graphs with diameter 2. For each of these classes either we prove NP-hardness or design a polynomial time algorithm.

Keywords

Balanced connected subgraph Trees Split graphs Chordal graphs Planar graphs Bipartite graphs NP-hard Color-balanced 
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Notes

Acknowledgement

We thank Florian Sikora for pointing out the connection with the Graph Motif problem.

References

  1. 1.
    Aichholzer, O., et al.: Balanced islands in two colored point sets in the plane. arXiv preprint arXiv:1510.01819 (2015)
  2. 2.
    Balachandran, N., Mathew, R., Mishra, T.K., Pal, S.P.: System of unbiased representatives for a collection of bicolorings. arXiv preprint arXiv:1704.07716 (2017)
  3. 3.
    Bereg, S., et al.: Balanced partitions of 3-colored geometric sets in the plane. Discret. Appl. Math. 181, 21–32 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Betzler, N., van Bevern, R., Fellows, M.R., Komusiewicz, C., Niedermeier, R.: Parameterized algorithmics for finding connected motifs in biological networks. IEEE/ACM Trans. Comput. Biol. Bioinform. 8(5), 1296–1308 (2011)CrossRefGoogle Scholar
  5. 5.
    Biniaz, A., Maheshwari, A., Smid, M.H.: Bottleneck bichromatic plane matching of points. In: CCCG (2014)Google Scholar
  6. 6.
    Böcker, S., Rasche, F., Steijger, T.: Annotating fragmentation patterns. In: Salzberg, S.L., Warnow, T. (eds.) WABI 2009. LNCS, vol. 5724, pp. 13–24. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-04241-6_2CrossRefGoogle Scholar
  7. 7.
    Bonnet, É., Sikora, F.: The graph motif problem parameterized by the structure of the input graph. Discret. Appl. Math. 231, 78–94 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Crowston, R., Gutin, G., Jones, M., Muciaccia, G.: Maximum balanced subgraph problem parameterized above lower bound. Theor. Comput. Sci. 513, 53–64 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Derhy, N., Picouleau, C.: Finding induced trees. Discret. Appl. Math. 157(17), 3552–3557 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dumitrescu, A., Kaye, R.: Matching colored points in the plane: some new results. Comput.Geom. 19(1), 69–85 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dumitrescu, A., Pach, J.: Partitioning colored point sets into monochromatic parts. Int. J. Comput. Geom. Appl. 12(05), 401–412 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    El-Kebir, M., Klau, G.W.: Solving the maximum-weight connected subgraph problem to optimality. CoRR abs/1409.5308 (2014)Google Scholar
  13. 13.
    Feige, U., Peleg, D., Kortsarz, G.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fellows, M.R., Fertin, G., Hermelin, D., Vialette, S.: Upper and lower bounds for finding connected motifs in vertex-colored graphs. J. Comput. Syst. Sci. 77(4), 799–811 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  16. 16.
    Johnson, D.S.: The NP-completeness column: an ongoing guide. J. Algorithms 6(1), 145–159 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane—a survey—. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry, vol. 25, pp. 551–570. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-642-55566-4_25CrossRefzbMATHGoogle Scholar
  18. 18.
    Kaneko, A., Kano, M., Watanabe, M.: Balancing colored points on a line by exchanging intervals. J. Inf. Process. 25, 551–553 (2017)Google Scholar
  19. 19.
    Kierstead, H.A., Trotter, W.T.: Colorful induced subgraphs. Discret. Math. 101(1–3), 165–169 (1992)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lacroix, V., Fernandes, C.G., Sagot, M.: Motif search in graphs: application to metabolic networks. IEEE/ACM Trans. Comput. Biol. Bioinform. 3(4), 360–368 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sujoy Bhore
    • 1
  • Sourav Chakraborty
    • 2
  • Satyabrata Jana
    • 2
  • Joseph S. B. Mitchell
    • 3
  • Supantha Pandit
    • 3
    Email author
  • Sasanka Roy
    • 2
  1. 1.Ben-Gurion UniversityBeer-ShevaIsrael
  2. 2.Indian Statistical InstituteKolkataIndia
  3. 3.Stony Brook UniversityStony BrookUSA

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