Abstract
For many constraint satisfaction problems, the algorithm which chooses a random assignment achieves the best possible approximation ratio. For instance, a simple random assignment for Max-E3-Sat allows 7/8-approximation and for every ε > 0 there is no polynomial-time (7/8 + ε)-approximation unless P=NP. Another example is the Permutation CSP of bounded arity. Given the expected fraction ρ of the constraints satisfied by a random assignment (i.e. permutation), there is no (ρ + ε)-approximation algorithm for every ε > 0, assuming the Unique Games Conjecture (UGC).
In this work, we consider the following parameterization of constraint satisfaction problems. Given a set of m constraints of constant arity, can we satisfy at least ρm + k constraint, where ρ is the expected fraction of constraints satisfied by a random assignment? Constraint Satisfaction Problems above Average have been posed in different forms in the literature [18,17]. We present a faster parameterized algorithm for deciding whether m/2 + k/2 equations can be simultaneously satisfied over \({\mathbb F}_2\). As a consequence, we obtain O(k)-variable bikernels for boolean CSPs of arity c for every fixed c, and for permutation CSPs of arity 3. This implies linear bikernels for many problems under the “above average” parameterization, such as Max-c-Sat, Set-Splitting, Betweenness and Max Acyclic Subgraph. As a result, all the parameterized problems we consider in this paper admit 2O(k)-time algorithms.
We also obtain non-trivial hybrid algorithms for every Max c-CSP: for every instance I, we can either approximate I beyond the random assignment threshold in polynomial time, or we can find an optimal solution to I in subexponential time.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: Solving MAX-r-SAT above a tight lower bound. Algorithmica (2010) (to appear)
Bodlaender, H., Fomin, F., Koster, A., Kratsch, D., Thilikos, D.: A note on exact algorithms for vertex ordering problems on graphs. Theory of Computing Systems, 1–13 (2010), doi:10.1007/s00224-011-9312-0
Bodlaender, H.L.: Kernelization: New Upper and Lower Bound Techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)
Calabro, C., Impagliazzo, R., Paturi, R.: A duality between clause width and clause density for sat. In: IEEE Conference on Computational Complexity, pp. 252–260 (2006)
Charikar, M., Guruswami, V., Manokaran, R.: Every permutation CSP of arity 3 is approximation resistant. In: 24th Annual IEEE Conference on Computational Complexity, CCC 2009, pp. 62–73 (July 2009)
Crowston, R., Gutin, G., Jones, M., Kim, E.J., Ruzsa, I.Z.: Systems of Linear Equations over \(\mathbb{F}_2\) and Problems Parameterized above Average. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 164–175. Springer, Heidelberg (2010)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Guruswami, V., Håstad, J., Manokaran, R., Raghavendra, P., Charikar, M.: Beating the random ordering is hard: Every ordering csp is approximation resistant. Electronic Colloquium on Computational Complexity (ECCC) 18, 27 (2011)
Guruswami, V., Manokaran, R., Raghavendra, P.: Beating the random ordering is hard: Inapproximability of maximum acyclic subgraph. In: FOCS, pp. 573–582 (2008)
Guruswami, V., Zhou, Y.: Approximating bounded occurrence ordering CSPs (2011) (manuscript)
Gutin, G., Kim, E.J., Mnich, M., Yeo, A.: Betweenness parameterized above tight lower bound. J. Comput. Syst. Sci. 76(8), 872–878 (2010)
Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: A probabilistic approach to problems parameterized above or below tight bounds. J. Comput. Syst. Sci. (2010) (to appear)
Gutin, G., van Iersel, L., Mnich, M., Yeo, A.: All Ternary Permutation Constraint Satisfaction Problems Parameterized above Average have Kernels with Quadratic Numbers of Variables. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 326–337. Springer, Heidelberg (2010)
Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)
Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the ACM symposium on Theory of Computing, pp. 767–775 (2002)
Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31(2), 335–354 (1999)
Mahajan, M., Raman, V., Sikdar, S.: Parameterizing above or below guaranteed values. J. Comput. System Sci. 75(2), 137–153 (2009)
Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)
O’Donnell, R.: Some topics in analysis of boolean functions. In: STOC, pp. 569–578 (2008)
Vassilevska, V., Williams, R., Woo, S.L.M.: Confronting hardness using a hybrid approach. In: SODA, pp. 1–10 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kim, E.J., Williams, R. (2012). Improved Parameterized Algorithms for above Average Constraint Satisfaction. In: Marx, D., Rossmanith, P. (eds) Parameterized and Exact Computation. IPEC 2011. Lecture Notes in Computer Science, vol 7112. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28050-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-28050-4_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28049-8
Online ISBN: 978-3-642-28050-4
eBook Packages: Computer ScienceComputer Science (R0)