Solving d-SAT via Backdoors to Small Treewidth

A backdoor set of a CNF formula is a set of variables such that fixing the truth values of the variables from this set moves the formula into a polynomial-time decidable class. In this work we obtain several algorithmic results for solving d-SAT, by exploiting backdoors to d-CNF formulas whose incidence graphs have small treewidth. For a CNF formula p and integer t, a strong backdoor set to treewidth t is a set of variables such that each possible partial assignment τ to this set reduces p to a formula whose incidence graph is of treewidth at most t. A weak backdoor set to treewidth t is a set of variables such that there is a partial assignment to this set that reduces p to a satisfiable formula of treewidth at most t. Our main contribution is an algorithm that, given a d-CNF formula p and an integer k, in time 2O(k)|p|, • either finds a satisfying assignment of p, or • reports correctly that p is not satisfiable, or • concludes correctly that p has no weak or strong backdoor set to treewidth t of size at most k. As a consequence of the above, we show that d-SAT parameterized by the size of a smallest weak/strong backdoor set to formulas of treewidth t, is fixed-parameter tractable. Prior to our work, such results were know only for the very special case of t = 1 (Gaspers and Szeider, ICALP 2012). Our result not only extends the previous work, it also improves the running time substantially. The running time of our algorithm is linear in the input size for every fixed k. Moreover, the exponential dependence on the parameter k is asymptotically optimal under Exponential Time Hypothesis (ETH). One of our main technical contributions is a linear time "protrusion replacer" improving over a O(nlog2 n)-time procedure of Fomin et al. (FOCS 2012). The new deterministic linear time protrusion replacer has several applications in kernelization and parameterized algorithms.

[1]  Marko Samer,et al.  Constraint satisfaction with bounded treewidth revisited , 2006, J. Comput. Syst. Sci..

[2]  Stefan Szeider.  Matched Formulas and Backdoor Sets , 2007, SAT.

[3]  Paul D. Seymour,et al.  Graph minors. V. Excluding a planar graph , 1986, J. Comb. Theory, Ser. B.

[4]  Dimitrios M. Thilikos,et al.  Explicit Linear Kernels via Dynamic Programming , 2013, SIAM J. Discret. Math..

[5]  Fedor V. Fomin,et al.  Bidimensionality and geometric graphs , 2011, SODA.

[6]  Fedor V. Fomin,et al.  Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[7]  Dimitrios M. Thilikos,et al.  (Meta) Kernelization , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[8]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[9]  Evgeny Dantsin,et al.  Worst-Case Upper Bounds , 2009, Handbook of Satisfiability.

[10]  Peter van Beek,et al.  Finding Small Backdoors in SAT Instances , 2011, Canadian Conference on AI.

[11]  Stefan Szeider,et al.  Backdoors to Acyclic SAT , 2012, ICALP.

[12]  Stephan Kreutzer,et al.  Computing excluded minors , 2008, SODA '08.

[13]  Bart Selman,et al.  Backdoors To Typical Case Complexity , 2003, IJCAI.

[14]  Rina Dechter,et al.  Tree Clustering for Constraint Networks , 1989, Artif. Intell..

[15]  Stefan Szeider,et al.  Strong Backdoors to Bounded Treewidth SAT , 2012, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[16]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[17]  Christophe Paul,et al.  Linear Kernels and Single-Exponential Algorithms Via Protrusion Decompositions , 2012, ICALP.

[18]  Russell Impagliazzo,et al.  Complexity of kSAT , 2007 .

[19]  Sharad Malik,et al.  Boolean satisfiability from theoretical hardness to practical success , 2009, Commun. ACM.

[20]  Jakub Gajarský,et al.  Kernelization Using Structural Parameters on Sparse Graph Classes , 2013, ESA.

[21]  Marko Samer,et al.  Constraint satisfaction with bounded treewidth revisited , 2010, J. Comput. Syst. Sci..

[22]  Dimitrios M. Thilikos,et al.  Bidimensionality and kernels , 2010, SODA '10.

[23]  Eugene C. Freuder.  A sufficient condition for backtrack-bounded search , 1985, JACM.

[24]  Fedor V. Fomin,et al.  The Multivariate Algorithmic Revolution and Beyond , 2012, Lecture Notes in Computer Science.

[25]  Stasys Jukna,et al.  Extremal Combinatorics - With Applications in Computer Science , 2001, Texts in Theoretical Computer Science. An EATCS Series.

[26]  B. Mohar,et al.  Graph Minors , 2009 .

[27]  Fedor V. Fomin,et al.  Treewidth computation and extremal combinatorics , 2008, Comb..

[28]  Chandra Chekuri,et al.  Polynomial bounds for the grid-minor theorem , 2013, J. ACM.

[29]  Marko Samer,et al.  Backdoor Sets of Quantified Boolean Formulas , 2008, Journal of Automated Reasoning.