, Volume 79, Issue 1, pp 230–250 | Cite as

Complexity and Approximability of Parameterized MAX-CSPs

  • Holger Dell
  • Eun Jung Kim
  • Michael Lampis
  • Valia Mitsou
  • Tobias Mömke


We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable–constraint incidence graph of the CSP instance. We consider Max-CSPs with the constraint types \({\text {AND}}\), \({\text {OR}}\), \({\text {PARITY}}\), and \({\text {MAJORITY}}\), and with various parameters k, and we attempt to fully classify them into the following three cases:
  1. 1.

    The exact optimum can be computed in \(\textsf {FPT}\) time.

  2. 2.

    It is Open image in new window -hard to compute the exact optimum, but there is a randomized \(\textsf {FPT}\) approximation scheme (\(\textsf {FPT\text {-}AS}\)), which computes a \((1{-}\epsilon )\)-approximation in time \(f(k,\epsilon ) \cdot {\text {poly}}(n)\).

  3. 3.

    There is no \(\textsf {FPT\text {-}AS}\) unless Open image in new window .

For the corresponding standard CSPs, we establish \(\textsf {FPT}\) versus Open image in new window -hardness results.


Constraint satisfaction problems Parameterized complexity Approximation Clique width Neighborhood diversity 


  1. 1.
    Alekhnovich, M., Razborov, A.A.: Satisfiability, branch-width and Tseitin tautologies. Comput. Complex. 20(4), 649–678 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amaldi, E., Kann, V.: The complexity and approximability of finding maximum feasible subsystems of linear relations. Theor. Comput. Sci. 147(1&2), 181–210 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amaldi, E., Kann, V.: On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems. Theor. Comput. Sci. 209(1–2), 237–260 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Austrin, P., Khot, S.: A characterization of approximation resistance for even k-partite csps. In: Kleinberg, R.D. (ed.) Innovations in Theoretical Computer Science, ITCS ’13, Berkeley, pp. 187–196. ACM (2013)Google Scholar
  5. 5.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Creignou, N.: A dichotomy theorem for maximum generalized satisfiability problems. J. Comput. Syst. Sci. 51(3), 511–522 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De, A., Mossel, E., Neeman, J.: Majority is stablest: discrete and sos. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) Symposium on Theory of Computing Conference, STOC’13, Palo Alto, pp. 477–486. ACM (2013)Google Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Elbassioni, K.M., Raman, R., Ray, S., Sitters, R.: On the approximability of the maximum feasible subsystem problem with 0/1-coefficients. In Mathieu, C. (ed.) Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, pp. 1210–1219. SIAM (2009)Google Scholar
  10. 10.
    Feldman, V., Guruswami, V., Raghavendra, P., Wu, Y.: Agnostic learning of monomials by halfspaces is hard. SIAM J. Comput. 41(6), 1558–1590 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ganian, R.: Twin-cover: beyond vertex cover in parameterized algorithmics. In: Marx, D., Rossmanith, P. (eds.) Parameterized and Exact Computation—6th International Symposium, IPEC 2011, Saarbrücken. Revised Selected Papers, vol. 7112, pp. 259–271 (2011)Google Scholar
  12. 12.
    Gaspers, S., Szeider, S.: Kernels for global constraints. In: Walsh, T. (ed.) IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, pp. 540–545. IJCAI/AAAI (2011)Google Scholar
  13. 13.
    Gaspers, S., Szeider, S.: Backdoors to acyclic SAT. In: Czumaj, A., Mehlhorn, K., Pitts, A.M., Wattenhofer, R. (eds.) Proceedings of Part I, Automata, Languages, and Programming—39th International Colloquium, ICALP 2012, Warwick. Lecture Notes in Computer Science, vol. 7391, pp. 363–374. Springer (2012)Google Scholar
  14. 14.
    Gaspers, S., Szeider, S.: Guarantees and limits of preprocessing in constraint satisfaction and reasoning. Artif. Intell. 216, 1–19 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grohe, M.: The structure of tractable constraint satisfaction problems. In: Kralovic, R., Urzyczyn, P. (eds.) Proceedings of MFCS 2006, Stará Lesná, Slovakia. Lecture Notes in Computer Science, vol. 4162, pp. 58–72. Springer (2006)Google Scholar
  16. 16.
    Gurski, F., Wanke, E.: The tree-width of clique-width bounded graphs without \(\mathit{K}_{{n, n}}\). In: Brandes, U., Wagner, D. (eds.) Proceedings of Graph-Theoretic Concepts in Computer Science, 26th International Workshop, WG 2000, Konstanz. Lecture Notes in Computer Science, vol. 1928, pp. 196–205. Springer (2000)Google Scholar
  17. 17.
    Guruswami, V., Raghavendra, P.: Hardness of learning halfspaces with noise. SIAM J. Comput. 39(2), 742–765 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Khanna, S., Sudan, M., Williamson, D.P.: A complete classification of the approximability of maximization problems derived from boolean constraint satisfaction. In: Leighton, F.T., Shor, P.W. (eds.) Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, pp. 11–20. ACM (1997)Google Scholar
  20. 20.
    Khot, S., Saket, R.: Approximating csps using LP relaxation. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) Proceedings of Part I, Automata, Languages, and Programming—42nd International Colloquium, ICALP 2015, Kyoto. Lecture Notes in Computer Science, vol. 9134, pp. 822–833. Springer (2015)Google Scholar
  21. 21.
    Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64(1), 19–37 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)CrossRefGoogle Scholar
  24. 24.
    Ordyniak, S., Paulusma, D., Szeider, S.: Satisfiability of acyclic and almost acyclic CNF formulas. Theor. Comput. Sci. 481, 85–99 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Paulusma, D., Slivovsky, F., Szeider, S.: Model counting for CNF formulas of bounded modular treewidth. In: Portier, N., Wilke, T. (eds.) STACS 2013, February 27–March 2, 2013, Kiel. LIPIcs, vol. 20, pp. 55–66. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)Google Scholar
  26. 26.
    Pichler, R., Rümmele, S., Szeider, S., Woltran, S.: Tractable answer-set programming with weight constraints: bounded treewidth is not enough. TPLP 14(2), 141–164 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Sæther, S.H., Telle, J.A., Vatshelle, M.: Solving maxsat and #sat on structured CNF formulas. In: Sinz, C., Egly, U. (eds.) Proceedings of SAT 2014—Vienna. Lecture Notes in Computer Science, vol. 8561, pp. 16–31. Springer (2014)Google Scholar
  28. 28.
    Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. J. Comput. Syst. Sci. 76(2), 103–114 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Lipton, R.J., Burkhard, W.A., Savitch, W.J., Friedman, E.P., Aho, A.V. (eds.) Proceedings of the 10th Annual ACM Symposium on Theory of Computing, San Diego, pp. 216–226. ACM (1978)Google Scholar
  30. 30.
    Slivovsky, F., Szeider, S.: Model counting for formulas of bounded clique-width. In: Cai, L., Cheng, S., Lam, T.W. (eds.) Proceedings of ISAAC 2013, Hong Kong, China. Lecture Notes in Computer Science, vol. 8283, pp. 677–687. Springer (2013)Google Scholar
  31. 31.
    Szeider, S.: On fixed-parameter tractable parameterizations of SAT. In: Giunchiglia, E., Tacchella, A. (eds.) Theory and Applications of Satisfiability Testing, 6th International Conference, SAT 2003. Santa Margherita Ligure, Selected Revised Papers. Lecture Notes in Computer Science, vol. 2919, pp. 188–202. Springer (2003)Google Scholar
  32. 32.
    Szeider, S.: Not so easy problems for tree decomposable graphs. CoRR, abs/1107.1177 (2011)Google Scholar
  33. 33.
    Szeider, S.: The parameterized complexity of constraint satisfaction and reasoning. In: Tompits, H., Abreu, S., Oetsch, J., Pührer, J., Seipel, D., Umeda, M., Wolf, A. (eds.) INAP 2011, and WLP 2011, Vienna, Revised Selected Papers. Lecture Notes in Computer Science, vol. 7773, pp. 27–37. Springer (2011)Google Scholar
  34. 34.
    Szeider, S.: The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optim. 8(1), 139–145 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Trevisan, L.: Inapproximability of combinatorial optimization problems. In: Paradigms of Combinatorial Optimization, 2nd edn, pp. 381–434 (2014). doi: 10.1002/9781119005353.ch13

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Saarland University and Cluster of ExcellenceSaarbrückenGermany
  2. 2.Université Paris DauphineParisFrance
  3. 3.SZTAKIHungarian Academy of SciencesBudapestHungary
  4. 4.Saarland UniversitySaarbrückenGermany

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