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Complexity and Approximability of Parameterized MAX-CSPs

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Abstract

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 -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 .

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

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Notes

  1. See the next section for a definition of incidence graphs.

  2. Akin to neighborhood diversity is the twin-cover number proposed in [11]. On bipartite graphs such as incidence graphs of CSPs, the twin-cover number is essentially the same as the vertex cover number:it differs only on a graph consisting of a single edge, in which the twin-cover number equals 0 while the vertex cover number is 1. Hence, we do not consider the twin-cover number separately as a structural parameter in this paper.

  3. We follow here the standard definition of \(\textsf {FPT\text {-}AS}\) given in [23].

  4. A CNF formula has bounded modular incidence treewidth if its incidence graph has bounded treewidth after merging variable/clause modules into a single vertex. Here, a variable/clause module is a set of vertices, corresponding to variables/clauses respectively, with same neighborhood outside of the set.In fact, the reduction in [25] constructs a formula whose incidence graph has small feedback vertex set after contracting modules.

References

  1. Alekhnovich, M., Razborov, A.A.: Satisfiability, branch-width and Tseitin tautologies. Comput. Complex. 20(4), 649–678 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Creignou, N.: A dichotomy theorem for maximum generalized satisfiability problems. J. Comput. Syst. Sci. 51(3), 511–522 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  8. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  14. Gaspers, S., Szeider, S.: Guarantees and limits of preprocessing in constraint satisfaction and reasoning. Artif. Intell. 216, 1–19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  17. Guruswami, V., Raghavendra, P.: Hardness of learning halfspaces with noise. SIAM J. Comput. 39(2), 742–765 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  21. Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64(1), 19–37 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)

    Article  Google Scholar 

  24. Ordyniak, S., Paulusma, D., Szeider, S.: Satisfiability of acyclic and almost acyclic CNF formulas. Theor. Comput. Sci. 481, 85–99 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  28. Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. J. Comput. Syst. Sci. 76(2), 103–114 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  32. Szeider, S.: Not so easy problems for tree decomposable graphs. CoRR, abs/1107.1177 (2011)

  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)

  34. Szeider, S.: The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optim. 8(1), 139–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Trevisan, L.: Inapproximability of combinatorial optimization problems. In: Paradigms of Combinatorial Optimization, 2nd edn, pp. 381–434 (2014). doi:10.1002/9781119005353.ch13

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Authors and Affiliations

  1. Saarland University and Cluster of Excellence, Saarbrücken, Germany

    Holger Dell

  2. Université Paris Dauphine, Paris, France

    Eun Jung Kim & Michael Lampis

  3. SZTAKI, Hungarian Academy of Sciences, Budapest, Hungary

    Valia Mitsou

  4. Saarland University, Saarbrücken, Germany

    Tobias Mömke

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  1. Holger Dell
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  2. Eun Jung Kim
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  3. Michael Lampis
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  4. Valia Mitsou
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Corresponding author

Correspondence to Holger Dell.

Additional information

Valia Mitsou: Supported by ERC Starting Grant PARAMTIGHT (No. 280152).

Tobias Mömke: This research is supported by Deutsche Forschungsgemeinschaft Grant BL511/10-1.

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Dell, H., Kim, E.J., Lampis, M. et al. Complexity and Approximability of Parameterized MAX-CSPs. Algorithmica 79, 230–250 (2017). https://doi.org/10.1007/s00453-017-0310-8

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