We present an O(1.3247ⁿ) algorithm for counting the number of satisfying assignments for instances of 2-SAT and an O(1.6894ⁿ) algorithm for instances of 3-SAT. This is an improvement compared to the previously best known algorithms running in O(1.381ⁿ) and O(1.739ⁿ) time, respectively.

We here present algorithms for counting models and max-weight models for 2SAT and 3SAT formulae. They use polynomial space and run in O(1.2561ⁿ) and O(1.6737ⁿ) time, respectively, where n is the number of variables. This is faster than the previously best algorithms for counting non-weighted models for 2SAT and 3SAT, which run in O(1.3247ⁿ) and O(1.6894ⁿ) time, respectively. In order to prove these time bounds, we develop new measures of formula complexity, allowing us to conveniently analyze the effects of certain factors with a large impact on the total running time. We also provide an algorithm for the restricted case of separable 2SAT formulae, with fast running times for well-studied input classes. For all three algorithms we present interesting applications, such as computing the permanent of sparse 0/1 matrices.

Partial clone theory has successfully been applied to study the complexity of the constraint satisfaction problem parameterized by a set of relations (CSP(G)). Lagerkvist & Wahlstroï¿œm (ISMVL 2014) however shows that the partial polymorphisms of G (?P?I(G)) cannot be finitely generated for finite, Boolean G if CSP(G) is NP-hard (assuming P?NP). In this paper we consider stronger closure operators than functional composition which can generate ?P?I(G) from a finite set of partial functions, a bounded base. Determining bounded bases for finite languages provides a complete characterization of their partial polymorphisms and we provide such bases for k-SAT and 1-in-k-SAT.

Two well-studied closure operators for relations are based on primitive positive (p.p.) definitions and quantifier free p.p. definitions. The latter do however have limited expressiveness and the corresponding lattice of strong partial clones is uncountable. We consider implementations allowing polynomially many existentially quantified variables and obtain a dichotomy for co-clones where such implementations are enough to implement any relation and prove (1) that all remaining coclones contain relations requiring a superpolynomial amount of quantified variables and (2) that the strong partial clones corresponding to two of these co-clones are of infinite order whenever the set of invariant relations can be finitely generated.

The topic of exact, exponential-time algorithms for NP-hard problems has received a lot of attention, particularly with the focus of producing algorithms with stronger theoretical guarantees, e.g. upper bounds on the running time on the form O(c^n) for some c. Better methods of analysis may have an impact not only on these bounds, but on the nature of the algorithms as well.

The most classic method of analysis of the running time of DPLL-style ("branching" or "backtracking") recursive algorithms consists of counting the number of variables that the algorithm removes at every step. Notable improvements include Kullmann's work on complexity measures, and Eppstein's work on solving multivariate recurrences through quasiconvex analysis. Still, one limitation that remains in Eppstein's framework is that it is difficult to introduce (non-trivial) restrictions on the applicability of a possible recursion.

We introduce two new kinds of complexity measures, representing two ways to add such restrictions on applicability to the analysis. In the first measure, the execution of the algorithm is viewed as moving between a finite set of states (such as the presence or absence of certain structures or properties), where the current state decides which branchings are applicable, and each branch of a branching contains information about the resultant state. In the second measure, it is instead the relative sizes of the modelled attributes (such as the average degree or other concepts of density) that controls the applicability of branchings.

We adapt both measures to Eppstein's framework, and use these tools to provide algorithms with stronger bounds for a number of problems. The problems we treat are satisfiability for sparse formulae, exact 3-satisfiability, 3-hitting set, and counting models for 2- and 3-satisfiability formulae, and in every case the bound we prove is stronger than previously known bounds.

We present an algorithm that decides the satisfiability of a CNF formula where every variable occurs at most k times in time O(2N(1&#x2212;c/(k+1)+O(1/k2)))" role="presentation" style="box-sizing: border-box; display: inline; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">O(2N(1−c/(k+1)+O(1/k2)))O(2N(1−c/(k+1)+O(1/k2))) for some c (that is, O(α N ) with α< 2 for every fixed k). For k ≤ 4, the results coincide with an earlier paper where we achieved running times of O(20.1736 N ) for k = 3 and O(20.3472N ) for k = 4 (for k ≤ 2, the problem is solvable in polynomial time). For k>4, these results are the best yet, with running times of O(20.4629N ) for k = 5 and O(20.5408N ) for k = 6. As a consequence of this, the same algorithm is shown to run in time O(20.0926L ) for a formula of length (i.e.total number of literals) L. The previously best bound in terms of L is O(20.1030L ). Our bound is also the best upper bound for an exact algorithm for a 3satformula with up to six occurrences per variable, and a 4sat formula with up to eight occurrences per variable.

We present two algorithms for the problem of finding a minimum transversal in a hypergraph of rank 3, also known as the 3-Hitting Set problem. This problem is a natural extension of the vertex cover problem for ordinary graphs. The first algorithm runs in time O(1.6538ⁿ) for a hypergraph with n vertices, and needs polynomial space. The second algorithm uses exponential space and runs in time O(1.6316ⁿ).

We present an algorithm that decides the satisfiability of a formula F on CNF form in time O(1.1279(d − 2)n), if F has at most d occurrences per variable or if F has an average of d occurrences per variable and no variable occurs only once. For d ≤ 4, this is better than previous results. This is the first published algorithm that is explicitly constructed to be efficient for cases with a low number of occurrences per variable. Previous algorithms that are applicable to this case exist, but as these are designed for other (more general, or simply different) cases, their performance guarantees for this case are weaker.