Algorithmics in Exponential Time

Exponential algorithms, i.e. algorithms of complexity O(cn) 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.

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