Max-Cut Parameterized Above the Edwards-Erdős Bound


We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut parameterized above the Edwards-Erdős bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size

$$ \frac{m}{2} + \frac{n-1}{4} + k $$

in time 2O(k)n 4, or decides that no such cut exists.

This answers a long-standing open question from parameterized complexity that has been posed a number of times over the past 15 years.

Our algorithm has asymptotically optimal running time, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computable kernel of polynomial size.

This is a preview of subscription content, access via your institution.


  1. 1.

    E.g. the definition in by Bandelt and Mulder [1] agrees with our definition of clique-forest, but the definition in Chap. 3 of Diestel’s book [10] has that every block graph is a tree.


  1. 1.

    Bandelt, H.J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory, Ser. B 41(2), 182–208 (1986)

    MATH  MathSciNet  Article  Google Scholar 

  2. 2.

    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Bollobás, B., Scott, A.: Better bounds for Max Cut. In: Contemporary Combinatorics. Bolyai Society Mathematical Studies, vol. 10, pp. 185–246 (2002)

    Google Scholar 

  4. 4.

    Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. J. Comput. Syst. Sci. 67(4), 789–807 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Charikar, M., Wirth, A.: Maximizing quadratic programs: extending Grothendieck’s inequality. In: Proc. FOCS 2004, pp. 54–60 (2004)

    Google Scholar 

  6. 6.

    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: Proc. FSTTCS 2011, pp. 229–240 (2011)

    Google Scholar 

  7. 7.

    Crowston, R., Gutin, G., Jones, M.: Directed acyclic subgraph problem parameterized above Poljak-Turzík bound. In: Proc. FSTTCS 2012, pp. 400–411 (2012)

    Google Scholar 

  8. 8.

    Crowston, R., Jones, M., Mnich, M.: Max-cut parameterized above the Edwards-Erdős bound. In: Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 7391, pp. 242–253 (2012)

    Google Scholar 

  9. 9.

    Crowston, R., Gutin, G., Jones, M., Muciaccia, G.: Maximum balanced subgraph problem parameterized above lower bound. In: Computing and Combinatorics. Lecture Notes in Computer Science, vol. 7936, pp. 434–445 (2013)

    Google Scholar 

  10. 10.

    Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer, Berlin (2010)

    Book  Google Scholar 

  11. 11.

    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, Berlin (1999)

    Book  Google Scholar 

  12. 12.

    Edwards, C.S.: Some extremal properties of bipartite subgraphs. Can. J. Math. 25, 475–485 (1973)

    MATH  Article  Google Scholar 

  13. 13.

    Edwards, C.S.: An improved lower bound for the number of edges in a largest bipartite subgraph. In: Recent Advances in Graph Theory, pp. 167–181 (1975)

    Google Scholar 

  14. 14.

    Erdős, P.: On some extremal problems in graph theory. Isr. J. Math. 3, 113–116 (1965)

    Article  Google Scholar 

  15. 15.

    Erdős, P.: On even subgraphs of graphs. Mat. Lapok 18, 283–288 (1967)

    MathSciNet  Google Scholar 

  16. 16.

    Erdős, P., Gyárfás, A., Kohayakawa, Y.: The size of the largest bipartite subgraphs. Discrete Math. 177, 267–271 (1997)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)

    Google Scholar 

  18. 18.

    Gutin, G., Yeo, A.: Note on maximal bisection above tight lower bound. Inf. Process. Lett. 110(21), 966–969 (2010)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Haglin, D.J., Venkatesan, S.M.: Approximation and intractability results for the maximum cut problem and its variants. IEEE Trans. Comput. 40(1), 110–113 (1991)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Hopcroft, J., Tarjan, R.: Efficient algorithms for graph manipulation. Commun. ACM 16(6), 372–378 (1973)

    Article  Google Scholar 

  21. 21.

    Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)

    MATH  MathSciNet  Article  Google Scholar 

  22. 22.

    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations. Proc. Symposium, IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y., 1972, pp. 85–103 (1972).

    Google Scholar 

  23. 23.

    Khot, S., O’Donnell, R.: SDP gaps and UGC-hardness for Max-Cut-Gain. Theory Comput. 5, 83–117 (2009)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. Tech. Rep. TR97-033, Electronic Colloquium on Computational Complexity (1997).

  25. 25.

    Mahajan, M., Raman, V., Sikdar, S.: Parameterizing above or below guaranteed values. J. Comput. Syst. Sci. 75(2), 137–153 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  26. 26.

    Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351(3), 394–406 (2006)

    MATH  MathSciNet  Article  Google Scholar 

  27. 27.

    Mnich, M., Zenklusen, R.: Bisections above tight lower bounds. In: Proc. WG 2012, pp. 184–193 (2012)

    Google Scholar 

  28. 28.

    Mnich, M., Philip, G., Saurabh, S., Suchý, O.: Beyond Max-Cut: λ-extendible properties parameterized above the Poljak-Turzík bound. In: Proc. FSTTCS 2012, pp. 412–423 (2012)

    Google Scholar 

  29. 29.

    Ngọc, N.V., Tuza, Zs.: Linear-time approximation algorithms for the max cut problem. Comb. Probab. Comput. 2(2), 201–210 (1993)

    Article  Google Scholar 

  30. 30.

    Poljak, S., Turzík, D.: A polynomial algorithm for constructing a large bipartite subgraph, with an application to a satisfiability problem. Can. J. Math. 34(3), 519–524 (1982)

    MATH  Article  Google Scholar 

  31. 31.

    Poljak, S., Tuza, Z.: Maximum cuts and large bipartite subgraphs. In: Combinatorial Optimization, New Brunswick, NJ, 1992–1993. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 20, pp. 181–244 (1995)

    Google Scholar 

  32. 32.

    Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theor. Comput. Sci. 351(3), 446–458 (2006)

    MATH  MathSciNet  Article  Google Scholar 

  33. 33.

    Sikdar, S.: Parameterizing from the Extremes: Feasible Parameterizations of some NP-optimization problems. Ph.D. thesis, The Institute of Mathematical Sciences, Chennai, India (2010)

Download references


We thank Tobias Friedrich and Gregory Gutin for help with the presentation of the results. We thank the anonymous reviewers for many useful comments and suggestions. Part of this research has been supported by an International Joint Grant from the Royal Society.

Author information



Corresponding author

Correspondence to Matthias Mnich.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Crowston, R., Jones, M. & Mnich, M. Max-Cut Parameterized Above the Edwards-Erdős Bound. Algorithmica 72, 734–757 (2015).

Download citation


  • Algorithms and data structures
  • Maximum cuts
  • Combinatorial bounds
  • Fixed-parameter tractability