Systems of Linear Equations over \(\mathbb{F}_2\) and Problems Parameterized above Average

  • Robert Crowston
  • Gregory Gutin
  • Mark Jones
  • Eun Jung Kim
  • Imre Z. Ruzsa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)


In the problem Max Lin, we are given a system Az = b of m linear equations with n variables over \(\mathbb{F}_2\) in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess.

Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC’06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least k, where k is the parameter.

It is not hard to see that we may assume that no two equations in Az = b have the same left-hand side and n = rank A. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: m ≤ 2 p(n) for an arbitrary fixed function p(n) = o(n). This result generalizes earlier results by Crowston et al. (arXiv:0911.5384) and Gutin et al. (Proc. IWPEC’09). We also prove that Max Lin AA is polynomial-time solvable for every fixed k and, moreover, Max Lin AA is in the parameterized complexity class W[P].

Max r-Lin AA is a special case of Max Lin AA, where each equation has at most r variables. In Max Exact r-SAT AA we are given a multiset of m clauses on n variables such that each clause has r variables and asked whether there is a truth assignment to the n variables that satisfies at least (1 − 2− r )m + k2− r clauses. Using our maximum excess results, we prove that for each fixed r ≥ 2, Max r-Lin AA and Max Exact r-SAT AA can be solved in time 2 O(k logk) + m O(1). This improves \(2^{O(k^2)}+m^{O(1)}\)-time algorithms for the two problems obtained by Gutin et al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively.

It is easy to see that maximization of arbitrary pseudo-boolean functions, i.e., functions \(f:\ \{-1,+1\}^n\rightarrow \mathbb{R}\), represented by their Fourier expansions is equivalent to solving Max Lin. Using our main maximum excess result, we obtain a tight lower bound on the maxima of pseudo-boolean functions.


Total Weight Boolean Function Fourier Expansion Parameterized Problem Computable Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: Solving MAX-r-SAT above a tight lower bound. Tech. Report arXiv:0907.4573. A priliminary version was published in Proc. ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 511–517 (2010),
  2. 2.
    Alon, N., Gutin, G., Krivelevich, M.: Algorithms with large domination ratio. J. Algorithms 50, 118–131 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boros, E., Hammer, P.L.: Pseudo-boolean optimization. Discrete Appl. Math. 123, 155–225 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Borwein, P.: Computational excursions in analysis and number theory. Springer, New York (2002)zbMATHGoogle Scholar
  5. 5.
    Crowston, R., Gutin, G., Jones, M.: Note on Max Lin-2 above average. Tech. Report, arXiv:0911.5384,
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (1999)Google Scholar
  7. 7.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  8. 8.
    Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: A probabilistic approach to problems parameterized above tight lower bound. In: Chen, J., Fomin, F.V. (eds.) Proc. IWPEC’09. LNCS, vol. 5917, pp. 234–245. Springer, Heidelberg (2009)Google Scholar
  9. 9.
    Jukna, S.: Extremal combinatorics: with applications in computer science. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  10. 10.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Håstad, J., Venkatesh, S.: On the advantage over a random assignment. Random Structures Algorithms 25(2), 117–149 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kleitman, D.J., Shearer, J.B., Sturtevant, D.: Intersection of k-element sets. Combinatorica 1, 381–384 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mahajan, M., Raman, V., Sikdar, S.: Parameterizing MAX SNP problems above guaranteed values. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 38–49. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Mahajan, M., Raman, V., Sikdar, S.: Parameterizing above or below guaranteed values. J. Computer System Sciences 75(2), 137–153 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford University Press, Oxford (2006)zbMATHCrossRefGoogle Scholar
  16. 16.
    O’Donnell, R.: Some topics in analysis of boolean functions. Technical report, ECCC Report TR08-055. Paper for an invited talk at STOC’08 (2008),
  17. 17.
    de Wolf, R.: A Brief Introduction to fourier analysis on the boolean cube. Theory Of Computing Library Graduate Surveys 1, 1–20 (2008), Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Robert Crowston
    • 1
  • Gregory Gutin
    • 1
  • Mark Jones
    • 1
  • Eun Jung Kim
    • 1
  • Imre Z. Ruzsa
    • 2
  1. 1.Department of Computer Science, Royal HollowayUniversity of LondonEghamUK
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

Personalised recommendations