Faster Algorithms for Weak Backdoors

A weak backdoor, or simply a backdoor, for a Boolean SAT formula F into a class of SAT formulae C is a partial truth assignment T such that F[T] is in C and satisfiability is preserved. The problem of finding a backdoor from class C1 into class C2, or WB(C1,C2), can be stated as follows: Given a formula F in C1, and a natural number k, determine whether there exists a backdoor for F into C2 assigning at most k variables. The class 0-Val contains all Boolean formulae with at least one negative literal in each clause. We design a new algorithm for WB(3CNF, 0-Val) by reducing it to a local search variant of 3-SAT. We show that our algorithm runs in time O*(2.562^k), improving on the previous state-of-the-art of O*(2.85^k). Here, the O* notation is a variant of the big-O notation that allows to omit polynomial factors in the input size. Next, we look at WB(3CNF, Null), where Null is the class consisting of the empty formula. This problem was known to have a trivial running time upper bound of O*(6^k) and can easily be solved in O*(3^k) time. We use a reduction to Conflict-Free-d-Hitting-Set to prove an upper bound of O*(2.2738^k), and also prove a lower bound of 2^o(k) assuming the Exponential Time Hypothesis. Finally, Horn is the class of formulae with at most one positive literal per clause. We improve the previous O*(4.54^k) running time for WB(3CNF, Horn) problem to O*(4.17^k), by exploiting the structure of the SAT instance to give a novel proof of the non-existence of the slowest cases after a slight restructuring of the branching priorities.