Document Open Access Logo

Fixed-Parameter Approximability of Boolean MinCSPs

Authors Édouard Bonnet, László Egri, Dániel Marx

PDF
Thumbnail PDF

File

LIPIcs.ESA.2016.18.pdf
  • Filesize: 1.18 MB
  • 18 pages

Document Identifiers

Author Details

Édouard Bonnet
László Egri
Dániel Marx

Cite AsGet BibTex

Édouard Bonnet, László Egri, and Dániel Marx. Fixed-Parameter Approximability of Boolean MinCSPs. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ESA.2016.18

Abstract

The minimum unsatisfiability version of a constraint satisfaction problem (CSP) asks for an assignment where the number of unsatisfied constraints is minimum possible, or equivalently, asks for a minimum-size set of constraints whose deletion makes the instance satisfiable. For a finite set Gamma of constraints, we denote by CSP(Gamma) the restriction of the problem where each constraint is from Gamma. The polynomial-time solvability and the polynomial-time approximability of CSP(Gamma) were fully characterized by [Khanna et al. SICOMP 2000]. Here we study the fixed-parameter (FP-) approximability of the problem: given an instance and an integer k, one has to find a solution of size at most g(k) in time f(k)n^{O(1)} if a solution of size at most k exists. We especially focus on the case of constant-factor FP-approximability. Our main result classifies each finite constraint language Gamma into one of three classes: (1) CSP(Gamma) has a constant-factor FP-approximation; (2) CSP(Gamma) has a (constant-factor) FP-approximation if and only if Nearest Codeword has a (constant-factor) FP-approximation; (3) CSP(Gamma) has no FP-approximation, unless FPT=W[P]. We show that problems in the second class do not have constant-factor FP-approximations if both the Exponential-Time Hypothesis (ETH) and the Linear PCP Conjecture (LPC) hold. We also show that such an approximation would imply the existence of an FP-approximation for the k-Densest Subgraph problem with ratio 1-epsilon for any epsilon>0.
Keywords
  • constraint satisfaction problems
  • approximability
  • fixed-parameter tractability

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Eric Allender, Michael Bauland, Neil Immerman, Henning Schnoor, and Heribert Vollmer. The complexity of satisfiability problems: Refining Schaefer’s theorem. J. Comput. Syst. Sci., 75(4):245-254, 2009. Google Scholar
  2. Sanjeev Arora, László Babai, Jacques Stern, and Z. Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci., 54(2):317-331, 1997. URL: http://dx.doi.org/10.1006/jcss.1997.1472.
  3. Arnab Bhattacharyya and Yuichi Yoshida. An algebraic characterization of testable boolean CSPs. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 123-134, 2013. Google Scholar
  4. V. G. Bodnarchuk, L. A. Kaluzhnin, V. N. Kotov, and B. A. Romov. Galois theory for Post algebras. Kibernetika, 5(3):1-10, 1969. Google Scholar
  5. Elmar Böhler, Steffen Reith, Henning Schnoor, and Heribert Vollmer. Bases for boolean co-clones. Inf. Process. Lett., 96(2):59-66, 2005. Google Scholar
  6. Édouard Bonnet, László Egri, and Dániel Marx. Fixed-parameter approximability of boolean minCSPs. CoRR, abs/1601.04935, 2016. URL: http://arxiv.org/abs/1601.04935.
  7. Édouard Bonnet, Bruno Escoffier, Eun Jung Kim, and Vangelis Th. Paschos. On subexponential and fpt-time inapproximability. Algorithmica, 71(3):541-565, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9889-1.
  8. Andrei A. Bulatov. A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM, 53(1):66-120, 2006. URL: http://dx.doi.org/10.1145/1120582.1120584.
  9. Andrei A. Bulatov. Complexity of conservative constraint satisfaction problems. ACM Trans. Comput. Log., 12(4):24, 2011. URL: http://dx.doi.org/10.1145/1970398.1970400.
  10. Andrei A. Bulatov. The complexity of the counting constraint satisfaction problem. J. ACM, 60(5):34, 2013. URL: http://dx.doi.org/10.1145/2528400.
  11. Andrei A. Bulatov, Martin E. Dyer, Leslie Ann Goldberg, Markus Jalsenius, Mark Jerrum, and David Richerby. The complexity of weighted and unweighted #CSP. J. Comput. Syst. Sci., 78(2):681-688, 2012. URL: http://dx.doi.org/10.1016/j.jcss.2011.12.002.
  12. Andrei A. Bulatov and Dániel Marx. The complexity of global cardinality constraints. Logical Methods in Computer Science, 6(4), 2010. URL: http://dx.doi.org/10.2168/LMCS-6(4:4)2010.
  13. Andrei A. Bulatov and Dániel Marx. Constraint satisfaction parameterized by solution size. SIAM J. Comput., 43(2):573-616, 2014. URL: http://dx.doi.org/10.1137/120882160.
  14. Liming Cai and Xiuzhen Huang. Fixed-parameter approximation: Conceptual framework and approximability results. Algorithmica, 57(2):398-412, 2010. URL: http://dx.doi.org/10.1007/s00453-008-9223-x.
  15. Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. Computational Complexity, 15(2):94-114, 2006. URL: http://dx.doi.org/10.1007/s00037-006-0210-9.
  16. Yijia Chen, Martin Grohe, and Magdalena Grüber. On parameterized approximability. In Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, pages 109-120, 2006. Google Scholar
  17. Yijia Chen, Martin Grohe, and Magdalena Grüber. On parameterized approximability. Electronic Colloquium on Computational Complexity (ECCC), 14(106), 2007. URL: http://eccc.hpi-web.de/eccc-reports/2007/TR07-106/index.html.
  18. Rajesh Hemant Chitnis, MohammadTaghi Hajiaghayi, and Guy Kortsarz. Fixed-parameter and approximation algorithms: A new look. In Parameterized and Exact Computation - 8th International Symposium, IPEC 2013, Sophia Antipolis, France, September 4-6, 2013, Revised Selected Papers, pages 110-122, 2013. URL: http://dx.doi.org/10.1007/978-3-319-03898-8_11.
  19. Nadia Creignou, Sanjeev Khanna, and Madhu Sudan. Complexity Classifications of Boolean Constraint Satisfaction Problems. Society for Industrial and Applied Mathematics, 2001. Google Scholar
  20. Nadia Creignou and Heribert Vollmer. Parameterized complexity of weighted satisfiability problems: Decision, enumeration, counting. Fundam. Inform., 136(4):297-316, 2015. Google Scholar
  21. Pierluigi Crescenzi and Alessandro Panconesi. Completeness in approximation classes. Inf. Comput., 93(2):241-262, 1991. URL: http://dx.doi.org/10.1016/0890-5401(91)90025-W.
  22. Robert Crowston, Gregory Gutin, Mark Jones, and Anders Yeo. Parameterized complexity of satisfying almost all linear equations over 𝔽₂. Theory Comput. Syst., 52(4):719-728, 2013. URL: http://dx.doi.org/10.1007/s00224-012-9415-2.
  23. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
  24. Víctor Dalmau and Andrei A. Krokhin. Robust satisfiability for csps: Hardness and algorithmic results. TOCT, 5(4):15, 2013. URL: http://dx.doi.org/10.1145/2540090.
  25. Víctor Dalmau, Andrei A. Krokhin, and Rajsekar Manokaran. Towards a characterization of constant-factor approximable min csps. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 847-857, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.58.
  26. Vladimir G. Deineko, Peter Jonsson, Mikael Klasson, and Andrei A. Krokhin. The approximability of MAXCSP with fixed-value constraints. J. ACM, 55(4), 2008. URL: http://dx.doi.org/10.1145/1391289.1391290.
  27. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4471-5559-1.
  28. Rodney G. Downey, Michael R. Fellows, Catherine McCartin, and Frances Rosamond. Parameterized approximation of dominating set problems. Inform. Process. Lett., 109(1):68-70, 2008. URL: http://dx.doi.org/10.1016/j.ipl.2008.09.017.
  29. J. Flum and M. Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin, 2006. Google Scholar
  30. D. Geiger. Closed systems of functions and predicates. Pacific Journal of Mathematics, 27:95-100, 1968. Google Scholar
  31. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
  32. Peter Jonsson, Mikael Klasson, and Andrei A. Krokhin. The approximability of three-valued MAXCSP. SIAM J. Comput., 35(6):1329-1349, 2006. URL: http://dx.doi.org/10.1137/S009753970444644X.
  33. Sanjeev Khanna, Madhu Sudan, Luca Trevisan, and David P. Williamson. The approximability of constraint satisfaction problems. SIAM J. Comput., 30(6):1863-1920, 2000. URL: http://dx.doi.org/10.1137/S0097539799349948.
  34. Vladimir Kolmogorov and Stanislav Zivny. The complexity of conservative valued CSPs. J. ACM, 60(2):10, 2013. URL: http://dx.doi.org/10.1145/2450142.2450146.
  35. Stefan Kratsch, Dániel Marx, and Magnus Wahlström. Parameterized complexity and kernelizability of MaxOnes and ExactOnes problems. In Mathematical Foundations of Computer Science 2010, 35th International Symposium, MFCS 2010, Brno, Czech Republic, August 23-27, 2010. Proceedings, pages 489-500, 2010. URL: http://dx.doi.org/10.1007/978-3-642-15155-2_43.
  36. Stefan Kratsch and Magnus Wahlström. Preprocessing of MinOnes problems: A dichotomy. In Automata, Languages and Programming, 37th International Colloquium, ICALP 2010, Bordeaux, France, July 6-10, 2010, Proceedings, Part I, pages 653-665, 2010. URL: http://dx.doi.org/10.1007/978-3-642-14165-2_55.
  37. Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Transactions on Algorithms, 11(2):15:1-15:31, 2014. URL: http://dx.doi.org/10.1145/2566616.
  38. Dániel Marx. Parameterized complexity of constraint satisfaction problems. Computational Complexity, 14(2):153-183, 2005. URL: http://dx.doi.org/10.1007/s00037-005-0195-9.
  39. Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60-78, 2008. Google Scholar
  40. Dániel Marx. Completely inapproximable monotone and antimonotone parameterized problems. J. Comput. Syst. Sci., 79(1):144-151, 2013. Google Scholar
  41. Dániel Marx and Igor Razgon. Constant ratio fixed-parameter approximation of the edge multicut problem. Inf. Process. Lett., 109(20):1161-1166, 2009. URL: http://dx.doi.org/10.1016/j.ipl.2009.07.016.
  42. Dana Moshkovitz and Ran Raz. Two-query PCP with subconstant error. J. ACM, 57(5), 2010. URL: http://dx.doi.org/10.1145/1754399.1754402.
  43. R. Pöschel and Kalužnin. Funktionen- und Relationenalgebren. Deutscher Verlag der Wissenschaften, Berlin, 1979. Google Scholar
  44. Emil L. Post. On the Two-Valued Iterative Systems of Mathematical Logic. Princeton University Press, 1941. Google Scholar
  45. Igor Razgon and Barry O'Sullivan. Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci., 75(8):435-450, 2009. Google Scholar
  46. Thomas J. Schaefer. The complexity of satisfiability problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, Calif., 1978), pages 216-226. ACM, New York, 1978. Google Scholar
  47. Johan Thapper and Stanislav Zivny. The complexity of finite-valued CSPs. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 695-704, 2013. URL: http://dx.doi.org/10.1145/2488608.2488697.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact