Worst-Case Upper Bounds

There are many algorithms for testing satisfiability — how to evaluate and compare them? It is common (but still disputable) to identify the efficiency of an algorithm with its worst-case complexity. From this point of view, asymptotic upper bounds on the worst-case running time and space is a criterion for evaluation and comparison of algorithms. In this chapter we survey ideas and techniques behind satisfiability algorithms with the currently best upper bounds. We also discuss some related questions: “easy” and “hard” cases of SAT, reducibility between various restricted cases of SAT, the possibility of solving SAT in subexponential time, etc. In Section 12.1 we define terminology and notation used throughout the chapter. Section 12.2 addresses the question of which special cases of SAT are polynomial-time tractable and which ones remain NP-complete. The first nontrivial upper bounds for testing satisfiability were obtained for algorithms that solve k-SAT; such algorithms also form the core of general SAT algorithms. Section 12.3 surveys the currently fastest algorithms for k-SAT. Section 12.4 shows how to use bounds for k-SAT to obtain the currently best bounds for SAT. Section 12.5 addresses structural questions like “what else happens if k-SAT is solvable in time 〈. . .〉?”. Finally, Section 12.6 summarizes the currently best bounds for the main cases of the satisfiability problem.