Algorithms for SAT and Upper Bounds on Their Complexity

The propositional satisfiability problem (SAT) is one of the most naturalNP-complete problems, and therefore its complexity is crucial for the computational complexity theory. Since SAT is NP-complete, it is unlikely that SAT can be solved in polynomial time. However, it is still important to understand how much time is required to solve SAT, even if this amount is exponential; an algorithm solving SAT in time, say, 2 would be quite useful for many applications, e.g., for contemporary circuit design problems. Research in SAT algorithms includes experimental study of their performance as well as theoretical study of their complexity. This survey is concerned with the theoretical aspect. We discuss some recent algorithms for SAT having nontrivial worst-case upper bounds on their complexity. We also discuss interesting related problems. For example, is it possible to find satisfying assignments for uniquely satisfiable Boolean formulas faster than to find satisfying assignments for arbitrary satisfiable Boolean formulas? This paper gives a more detailed review of the existing SAT algorithms having the best current worst-case upper bounds. We survey families of such algorithms and mention some clarifying illustrations and open problems. This paper contains the following new results: we simplify a derandomization of the satisfiability coding lemma [38], give two new proofs of the Valiant–Vazirani lemma [46], and prove some results concerning walk algorithms for SAT.