Complexity of SAT Problems, Clone Theory and the Exponential Time Hypothesis

The construction of exact exponential-time algorithms for NP-complete problems has for some time been a very active research area. Unfortunately, there is a lack of general methods for studying and comparing the time complexity of algorithms for such problems. We propose such a method based on clone theory and demonstrate it on the SAT problem. Schaefer has completely classified the complexity of SAT with respect to the set of allowed relations and proved that this parameterized problem exhibits a dichotomy: it is either in P or is NP-complete. We show that there is a certain partial order on the NP-complete SAT problems with a close connection to their worst-case time complexities; if a problem SAT(S) is below a problem SAT(S') in this partial order, then SAT(S') cannot be solved strictly faster than SAT(S). By using this order, we identify a relation R such that SAT({R}) is the computationally easiest NP-complete SAT(S') problem. This result may be interesting when investigating the borderline between P and NP since one appealing way of studying this borderline is to identify problems that, in some sense, are situated close to it (such as a 'very hard' problem in P or a 'very easy' NP-complete problem). We strengthen the result by showing that SAT({R})-2 (i.e. SAT({R}) restricted to instances where no variable appears more than twice) is NP-complete, too. This is in contrast to, for example, 1-in-3-SAT (or even CNF-SAT), which is in P under the same restriction. We then relate SAT({R})-2 to the exponential-time hypothesis (ETH) and show that ETH holds if and only if SAT({R})-2 is not sub-exponential. This constitutes a strong connection between ETH and the SAT problem under both severe relational and severe structural restrictions, and it may thus serve as a tool for studying the borderline between sub-exponential and exponential problems. In the process, we also prove a stronger version of Impagliazzo et al.'s sparsification lemma for k-SAT; namely that all finite Boolean constraint languages S and S' such that SAT(·) is NP-complete can be sparsified into each other. This should be compared with Santhanam and Srinivasan's recent negative result which states that the same does not hold for all infinite Boolean constraint languages.

[1]  Hans L. Bodlaender,et al.  Partition Into Triangles on Bounded Degree Graphs , 2012, Theory of Computing Systems.

[2]  Russell Impagliazzo,et al.  Which Problems Have Strongly Exponential Complexity? , 2001, J. Comput. Syst. Sci..

[3]  Peter Jeavons,et al.  Classifying the Complexity of Constraints Using Finite Algebras , 2005, SIAM J. Comput..

[4]  D. Lau,et al.  Function Algebras on Finite Sets: Basic Course on Many-Valued Logic and Clone Theory (Springer Monographs in Mathematics) , 2006 .

[5]  Uwe Schöning,et al.  A Low and a High Hierarchy within NP , 1983, J. Comput. Syst. Sci..

[6]  Peter Jeavons,et al.  On the Algebraic Structure of Combinatorial Problems , 1998, Theor. Comput. Sci..

[7]  Víctor Dalmau,et al.  Generalized Satisfability with Limited Occurrences per Variable: A Study through Delta-Matroid Parity , 2003, MFCS.

[8]  D. Geiger.  CLOSED SYSTEMS OF FUNCTIONS AND PREDICATES , 1968 .

[9]  L. A. Kaluzhnin,et al.  Galois theory for Post algebras. II , 1969 .

[10]  Phokion G. Kolaitis,et al.  Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics , 2002, CP.

[11]  Cristopher Moore,et al.  Hard Tiling Problems with Simple Tiles , 2001, Discret. Comput. Geom..

[12]  Lucien Haddad.  Infinite Chains of Partial Clones Containing All Selfdual Monotonic Partial Functions , 2012, J. Multiple Valued Log. Soft Comput..

[13]  Gabriel Istrate.  Looking for a Version of Schaefer''s Dichotomy Theorem When Each Variable Occurs at Most Twice , 1997 .

[14]  Rahul Santhanam,et al.  On the Limits of Sparsification , 2012, ICALP.

[15]  L. A. Kaluzhnin,et al.  Galois theory for post algebras. I , 1969 .

[16]  Ágnes Szendrei,et al.  Clones in universal algebra , 1986 .

[17]  Dániel Marx,et al.  Lower bounds based on the Exponential Time Hypothesis , 2011, Bull. EATCS.

[18]  Gustav Nordh,et al.  Frozen Boolean Partial Co-clones , 2009, 2009 39th International Symposium on Multiple-Valued Logic.

[19]  Peter Jonsson,et al.  Hard constraint satisfaction problems have hard gaps at location 1 , 2007, Theor. Comput. Sci..

[20]  Rahul Santhanam,et al.  Explorer On the Limits of Sparsification , 2012 .

[21]  Manuel Bodirsky,et al.  Non-dichotomies in Constraint Satisfaction Complexity , 2008, ICALP.

[22]  Yoshio Okamoto,et al.  On Problems as Hard as CNFSAT , 2011, ArXiv.

[23]  Dominik Scheder,et al.  A full derandomization of schöning's k-SAT algorithm , 2010, STOC.

[24]  Emil L. Post.  The two-valued iterative systems of mathematical logic , 1942 .

[25]  Martin Grohe,et al.  The complexity of homomorphism and constraint satisfaction problems seen from the other side , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[26]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[27]  Russell Impagliazzo,et al.  On the Complexity of k-SAT , 2001, J. Comput. Syst. Sci..

[28]  B. A. Romov.  The algebras of partial functions and their invariants , 1981 .

[29]  Magnus Wahlström,et al.  Algorithms, measures and upper bounds for satisfiability and related problems , 2007 .

[30]  Dániel Marx.  Can you beat treewidth? , 2007, FOCS.

[31]  Henning Schnoor,et al.  New Algebraic Tools for Constraint Satisfaction , 2006, Complexity of Constraints.

[32]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[33]  Gerhard J. Woeginger,et al.  Exact Algorithms for NP-Hard Problems: A Survey , 2001, Combinatorial Optimization.