We prove the worst-case upper bound 1:5045 n for the time complexity of 3-SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2- and 3-clauses, called "blocked clauses", generalizing the extension rule of "Extended Resolution." Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3-clause-sets and can be applied also to arbitrary trees. Keywords: 3-SAT, worst-case upper bounds, analysis of algorithms, Extended Resolution, blocked clauses, generalized autarkness. 1 Introduction In this paper we study the exponential part of time complexity for 3-SAT decision and prove the worst-case upper bound 1:5044:: n for n the number of variables in the input formula, using new algorithmic methods as well as new methods for the analysis. These methods also deepen the already existing approaches in a systematically manner. The following results...

Abstract. We show that satisfiability of formulas in k-CNF can be decided deterministically in time close to (2k/(k + 1)) n, where n is the number of variables in the input formula. This is the best known worstcase upper bound for deterministic k-SAT algorithms. Our algorithm can be viewed as a derandomized version of Schöning’s probabilistic algorithm presented in [15]. The key point of our algorithm is the use of covering codes together with local search. Compared to other “weakly exponential ” algorithms, our algorithm is technically quite simple. We also show how to improve the bound above by moderate technical effort. For 3-SAT the improved bound is 1.481 n. 1

We describe exact algorithms that provide new upper bounds for the Maximum Satisfiability problem (MaxSat). We prove that MaxSat can be solved in time O(|F | · 1.3972 K), where |F | is the length of a formula F in conjunctive normal form and K is the number of clauses in F. We also prove the time bounds O(|F | · 1.3995 k), where k is the maximum number of satisfiable clauses, and O((1.1279) |F | ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K). An exponential time approximation algorithm by Dantsin et al. uses an exact algorithm for MaxSat as a building block and is therefore also improved.

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 dpllstyle (“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 give an algorithm for counting the number of max-weight solutions to a 2SAT formula, and improve the bound on its running time to O (1.2377 n). The main source of the improvement is a refinement of the method of analysis, where we extend the concept of compound (piecewise linear) measures to multivariate measures, also allowing the optimal parameters for the measure to be found automatically. This method extension should be of independent interest.

Counting the number of solutions to CSP instances has applications in several areas, ranging from statistical physics to artificial intelligence. We give an algorithm for counting the number of solutions to binary CSPs, which works by transforming the problem into a number of 2-sat instances, where the total number of solutions to these instances is the same as those of the original problem. The algorithm consists of two main cases, depending on whether the domain size d is even, in which case the algorithm runs in O(1:3247 ) time, or odd, in which case it runs in O(1:3247 ) if d = 4 \Delta k + 1, and O(1:3247 ) if d = 4 \Delta k + 3. We also give an algorithm for counting the number of possible 3-colourings of a given graph, which runs in O(1:8171 ), an improvement over our general algorithm gained by using problem specific knowledge.

We present an adaptation of transfer matrix method for signed grids, grid-cylinders and grid-tori. We use this adaptation to count the number of satisfying assignments of Boolean Formulas in 2-CNF whose corresponding associated graph has such grid topologies. 1

We present a new deterministic algorithm for the #3-SAT problem, based on the DPLL strategy. It uses a new approach for counting models of instances with low density. This allows us to assume the adding of more 2-clauses than in previous algorithms. The algorithm achieves a running time of O(1.6423 n) in the worst case which improves the current best bound of O(1.6737 n) by Dahllöf et al. 1