Derandomization of $\boldsymbol{k}$-SAT Algorithm

Years and Authors of Summarized Original Work

• 2011; Moser, Scheder

Problem Definition

Satisfiability is the central NP-complete problem. Given a Boolean formula in conjunctive normal form, for example, $\left (x \vee \bar{ y} \vee z\right ) \wedge \left (\bar{x} \vee \bar{ z}\right )\wedge \ldots$, decide whether there is a satisfying assignment. An important subclass is k-SAT, where the input is restricted to k-CNF formulas: CNF formulas in which every clause has at most k literals. In 1999, Uwe Schöning [6] gave an extremely simple randomized algorithm for k-SAT of running time

$$\displaystyle{\left (\frac{2(k - 1)} {k} \right )^{n}\,\mathrm{poly}(n).}$$

In particular this solves 3-SAT in time O^∗(1. 334ⁿ), 4-SAT in O^∗(1. 5ⁿ) for 4-SAT, and so on (we use O^∗ to suppress polynomial factors in n). Several authors have attempted to derandomize Schöning’s algorithm, albeit at the cost of a greater running time: an algorithm of Dantsin, Goerdt, Hirsch, Kannan, Kleinberg, Papadimitriou,...