Tighter hard instances for PPSZ

We construct uniquely satisfiable k-CNF formulas that are hard for the PPSZ algorithm, the currently best known algorithm solving k-SAT. This algorithm tries to generate a satisfying assignment by picking a random variable at a time and attempting to derive its value using some inference heuristic and otherwise assigning a random value. The "weak PPSZ" checks all subformulas of a given size to derive a value and the "strong PPSZ" runs resolution with width bounded by some given function. Firstly, we construct graph-instances on which "weak PPSZ" has savings of at most (2 + epsilon)/k; the saving of an algorithm on an input formula with n variables is the largest gamma such that the algorithm succeeds (i.e. finds a satisfying assignment) with probability at least 2^{- (1 - gamma) n}. Since PPSZ (both weak and strong) is known to have savings of at least (pi^2 + o(1))/6k, this is optimal up to the constant factor. In particular, for k=3, our upper bound is 2^{0.333... n}, which is fairly close to the lower bound 2^{0.386... n} of Hertli [SIAM J. Comput.'14]. We also construct instances based on linear systems over F_2 for which strong PPSZ has savings of at most O(log(k)/k). This is only a log(k) factor away from the optimal bound. Our constructions improve previous savings upper bound of O((log^2(k))/k) due to Chen et al. [SODA'13].