Maximum Balanced Subgraph Problem Parameterized above Lower Bound

  • Robert Crowston
  • Gregory Gutin
  • Mark Jones
  • Gabriele Muciaccia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7936)


We consider graphs without loops or parallel edges in which every edge is assigned + or −. Such a signed graph is balanced if its vertex set can be partitioned into parts V1 and V2 such that all edges between vertices in the same part have sign + and all edges between vertices of different parts have sign − (one of the parts may be empty). It is well-known that every connected signed graph with n vertices and m edges has a balanced subgraph with at least \(\frac{m}{2} + \frac{n-1}{4}\) edges and this bound is tight. We consider the following parameterized problem: given a connected signed graph G with n vertices and m edges, decide whether G has a balanced subgraph with at least \(\frac{m}{2} + \frac{n-1}{4}+\frac{k}{4}\) edges, where k is the parameter.

We obtain an algorithm for the problem of runtime 8k(kn)O(1). We also prove that for each instance (G,k) of the problem, in polynomial time, we can either solve (G,k) or produce an equivalent instance (G′,k′) such that k′ ≤ k and |V(G′)| = O(k3). Our first result generalizes a result of Crowston, Jones and Mnich (ICALP 2012) on the corresponding parameterization of Max Cut (when every edge of G has sign −). Our second result generalizes and significantly improves the corresponding result of Crowston, Jones and Mnich for MaxCut: they showed that |V(G′)| = O(k5).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chiang, C., Kahng, A.B., Sinha, S., Xu, X., Zelikovsky, A.Z.: Fast and efficient bright-field AAPSM conflict detection and correction. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 26(1), 11–126 (2007)CrossRefGoogle Scholar
  2. 2.
    Crowston, R., Fellows, M., Gutin, G., Jones, M., Rosamond, F., Thomassé, S., Yeo, A.: Simultaneously Satisfying Linear Equations Over \(\mathbb{F}_2\): MaxLin2 and Max-r-Lin2 Parameterized Above Average. In: FSTTCS 2011. LIPICS, vol. 13, pp. 229–240 (2011)Google Scholar
  3. 3.
    Crowston, R., Gutin, G., Jones, M.: Directed Acyclic Subgraph Problem Parameterized above the Poljak-Turzík Bound. In: FSTTCS 2012. LIPICS, vol. 18, pp. 400–411 (2012)Google Scholar
  4. 4.
    Crowston, R., Gutin, G., Jones, M., Muciaccia, G.: Maximum Balanced Subgraph Problem Parameterized Above Lower Bound. arXiv:1212.6848Google Scholar
  5. 5.
    Crowston, R., Jones, M., Mnich, M.: Max-Cut Parameterized above the Edwards-Erdős Bound. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 242–253. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    DasGupta, B., Enciso, G.A., Sontag, E.D., Zhang, Y.: Algorithmic and complexity results for decompositions of biological networks into monotone subsystems. Biosystems 90(1), 161–178 (2007)CrossRefGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  9. 9.
    Gülpınar, N., Gutin, G., Mitra, G., Zverovitch, A.: Extracting Pure Network Submatrices in Linear Programs Using Signed Graphs. Discrete Applied Mathematics 137, 359–372 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gutin, G., Zverovitch, A.: Extracting pure network submatrices in linear programs using signed graphs, Part 2. Communications of DQM 6, 58–65 (2003)Google Scholar
  11. 11.
    Harary, F.: On the notion of balance of a signed graph. Michigan Math. J. 2(2), 143–146 (1953)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hüffner, F., Betzler, N., Niedermeier, R.: Optimal edge deletions for signed graph balancing. In: Demetrescu, C. (ed.) WEA 2007. LNCS, vol. 4525, pp. 297–310. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Mnich, M., Philip, G., Saurabh, S., Suchý, O.: Beyond Max-Cut: λ-Extendible Properties Parameterized Above the Poljak-Turzík Bound. In: FSTTCS 2012. LIPICS, vol. 18, pp. 412–423 (2012)Google Scholar
  14. 14.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms, Oxford UP (2006)Google Scholar
  15. 15.
    Poljak, S., Turzík, D.: A polynomial time heuristic for certain subgraph optimization problems with guaranteed worst case bound. Discrete Mathematics 58(1), 99–104 (1986)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theor. Comput. Sci. 351(3), 446–458 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Zaslavsky, T.: Bibliography of signed and gain graphs. Electronic Journal of Combinatorics, DS8 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Robert Crowston
    • 1
  • Gregory Gutin
    • 1
  • Mark Jones
    • 1
  • Gabriele Muciaccia
    • 1
  1. 1.Royal Holloway, University of LondonUK

Personalised recommendations