Algorithmics in Exponential Time

Exponential algorithms, i.e. algorithms of complexity O(c) for some c > 1, seem to be unavoidable in the case of NP-complete problems (unless P=NP), especially if the problem in question needs to be solved exactly and not approximately. If the constant c is close to 1 such algorithms have practical importance. Deterministic algorithms of exponential complexity usually involve some kind of backtracking. The analysis of such backtracking algorithms in terms of solving recurrence equations is quite well understood. The purpose of the current paper is to show cases in which the constant c could be significantly reduced, and to point out that there are some randomized exponential-time algorithms which use randomization in some new ways. Most of our examples refer to the 3-SAT problem, i.e. the problem of determining satisfiability of formulas in conjunctive normal form with at most 3 literals per clause. 1 Why Exponential-Time Algorithms? Unless P=NP, exponential (or at least non-polynomial) algorithms are unavoidable for NP-complete problems such as SAT or 3-SAT. This is especially so if the problem in question needs to be solved or decided exactly, and when there is no use of any type of approximation algorithm. Having accepted that we have to deal with exponential-time algorithms only, it makes sense to improve the relative efficiency of our algorithms by reducing the value of the base constant c in the exponential time bound O(c). New algorithms which are able to reduce the constant c can mean an tremendous improvement. Moving from an algorithm with complexity O(c) to another, better one, with complexity O(d) where d = √ c means that, within the same given time limit, the input size that can be solved by the new algorithm, doubles. In concrete terms, improving the brute-force algorithm for 3-SAT of complexity O(2) (where n is the number of Boolean variables) to another one of complexity O(1.324) (as it was indeed the case within the last years [14, 17, 10]). This improvement allows to increase the number of tractable Boolean variables by a factor of more than 2.4 . Another motivating example are SAT-solvers which are extremely useful general-purpose programs which are operating in worst-case exponential-time. V. Diekert and B. Durand (Eds.): STACS 2005, LNCS 3404, pp. 36–43, 2005. c © Springer-Verlag Berlin Heidelberg 2005 Algorithmics in Exponential Time 37