Faster Randomized Branching Algorithms for $r$-SAT

The problem of determining if an r-CNF boolean formula F over n variables is satisifiable reduces to the problem of determining if F has a satisfying assignment with a Hamming distance of at most d from a fixed assignment α [DGH02]. This problem is also a very important subproblem in Schöning’s local search algorithm for r-SAT [Sch99]. While Schöning described a randomized algorithm solves this subproblem in O((r − 1)) time, Dantsin et al. [DGH02] presented a deterministic branching algorithm with O(r) running time. In this paper we present a simple randomized branching algorithm that runs in time O(( r+1 2 ) d ). As a consequence we get a randomized algorithm for r-SAT that runs inO∗(( r+3 ) n ) time. This algorithm matches the running time of Schöning’s algorithm for 3-SAT and is an improvement over Schöning’s algorithm for all r > 4. For r-uniform hitting set parameterized by solution size k, we describe a randomized FPT algorithm with a running time of O(( r+1 2 ) k ). For the above LP guarantee parameterization of vertex cover, we have a randomized FPT algorithm to find a vertex cover of size k in a running time of O(2.25 ∗ ), where vc is the LP optimum of the natural LP relaxation of vertex cover. In both the cases, these randomized algorithms have a better running time than the current best deterministic algorithms, though they do fail with very small probability.