Algorithms, measures and upper bounds for satisfiability and related problems

Abstract: 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.