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We construct uniquely satisfiable $k$-CNF formulas that are hard for thealgorithm PPSZ. Firstly, we construct graph-instances on which "weak PPSZ" hassavings of at most $(2 + \epsilon) / k$; the saving of an algorithm on an inputformula with $n$ variables is the largest $\gamma$ such that the algorithmsucceeds (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 ofat least $\frac{\pi^2 + o(1)}{6k}$, this is optimal up to the constant factor.In particular, for $k=3$, our upper bound is $2^{0.333\dots n}$, which isfairly close to the lower bound $2^{0.386\dots n}$ of Hertli [SIAM J.Comput.'14]. We also construct instances based on linear systems over$\mathbb{F}_2$ for which strong PPSZ has savings of at most$O\left(\frac{\log(k)}{k}\right)$. This is only a $\log(k)$ factor away fromthe optimal bound. Our constructions improve previous savings upper bound of$O\left(\frac{\log^2(k)}{k}\right)$ due to Chen et al. [SODA'13].

Tighter Hard Instances for PPSZ