Improving exact exponential-time algorithms for NP-complete problems is an expanding research area. Unfortunately, general methods for comparing the complexity of such problems is sorely lacking. In this article we study the complexity of SAT(S) with reductions increasing the amount of variables by a constant (CV-reductions) or a constant factor (LV-reductions). Using clone theory we obtain a partial order ≤ on languages such that SAT(S) is CV-reducible to SAT(S) if S ≤ S. With this ordering we identify the computationally easiest NP-complete SAT(S) problem (SAT({R})), which is strictly easier than 1-in-3-SAT. We determine many other languages in ≤ and bound their complexity in relation to SAT({R}). Using LV-reductions we prove that the exponential-time hypothesis is false if and only if all SAT(S) problems are subexponential. This is extended to cover degree-bounded SAT(S) problems. Hence, using clone theory, we obtain a solid understanding of the complexity of SAT(S) with CV-and LV-reductions.

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 classied 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

Universal algebra and clone theory have proven to be a useful tool in the study of constraint satisfaction problems since the complexity, up to logspace reductions, is determined by the set of polymorphisms of the constraint language. For classifications where primitive positive definitions are unsuitable, such as sizepreserving reductions, weaker closure operations may be necessary. In this article we consider strong partial clones which can be seen as a more fine-grained framework than Post's lattice where each clone splits into an interval of strong partial clones. We investigate these intervals and give simple relational descriptions, weak bases, of the largest elements. The weak bases have a highly regular form and are in many cases easily relatable to the smallest members in the intervals, which suggests that the lattice of strong partial clones is considerably simpler than the full lattice of partial clones.

The self-archived version of this journal article is available at Linköping University Institutional Repository (DiVA): http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-139605 N.B.: When citing this work, cite the original publication. Lagerkvist, V., Wahlström, M., (2017) ed variables, and say that a co-clone closed under such denitions is polynomially closed, and otherwise superpolynomially closed. We investigate properties of polynomially closed co-clones and prove that if the corresponding clone contains a k-ary near-unanimity operation for k ≥ 3 then the co-clone is polynomially closed, and if the clone does not contain a k-edge operation for any k ≥ 2, then the co-clone is superpolynomially closed. For the Boolean domain we strengthen these results and prove a complete dichotomy theorem separating polynomially closed co-clones from superpolynomially closed co-clones. Using these results, we then proceed to investigate properties of strong partial clones corresponding to superpolynomially closed co-clones. We prove that if Γ is a nite set of relations over an arbitrary nite domain such that the clone corresponding to Γ is essentially unary, then the strong partial clone corresponding to Γ is of innite order and cannot be generated by a nite set of partial functions.

A kernelization algorithm for a computational problem is a procedure which compresses an instance into an equivalent instance whose size is bounded with respect to a complexity parameter. For the Boolean satisfiability problem (SAT), and the constraint satisfaction problem (CSP), there exist many results concerning upper and lower bounds for kernelizability of specific problems, but it is safe to say that we lack general methods to determine whether a given SAT problem admits a kernel of a particular size. This could be contrasted to the currently flourishing research program of determining the classical complexity of finite-domain CSP problems, where almost all non-trivial tractable classes have been identified with the help of algebraic properties. In this paper, we take an algebraic approach to the problem of characterizing the kernelization limits of NP-hard SAT and CSP problems, parameterized by the number of variables. Our main focus is on problems admitting linear kernels, as has, somewhat surprisingly, previously been shown to exist. We show that a CSP problem has a kernel with O(n) constraints if it can be embedded (via a domain extension) into a CSP problem which is preserved by a Maltsev operation. We also study extensions of this towards SAT and CSP problems with kernels with O(n c ) constraints, c > 1, based on embeddings into CSP problems preserved by a k-edge operation, k ≤ c + 1. These results follow via a variant of the celebrated few subpowers algorithm. In the complementary direction, we give indication that the Maltsev condition might be a complete characterization of SAT problems with linear kernels, by showing that an algebraic condition that is shared by all problems with a Maltsev embedding is also necessary for the existence of a linear kernel unless NP ⊆ co-NP/poly.