Solving d-SAT via backdoors to small Treewidth

A backdoor set of a CNF formula is a set of variables such that fixing the truth values of the variables from this set moves the formula into a polynomial-time decidable class. In this work we obtain several algorithmic results for solving d-SAT, by exploiting backdoors to d-CNF formulas whose incidence graphs have small treewidth. For a CNF formula F and integer t, a strong backdoor set to treewidth t is a set of variables such that each possible partial assignment τ to this set reduces F to a formula whose incidence graph is of treewidth at most t. A weak backdoor set to treewidth t is a set of variables such that there is a partial assignment to this set that reduces φ to a satisfiable formula of treewidth at most t. Our main contribution is an algorithm that, given a d-CNF formula F and an integer k, in time 2^{O(k)}|F|, • either finds a satisfying assignment of F, or • reports correctly that F is not satisfiable, or • concludes correctly that F has no weak or strong backdoor set to treewidth t of size at most k. As a consequence of the above, we show that d-SAT parameterized by the size of a smallest weak/strong backdoor set to formulas of treewidth t, is fixed-parameter tractable