Tighter Hard Instances for PPSZ

TL;DR

This paper constructs specific hard instances of k-CNF formulas that challenge the PPSZ algorithm, establishing near-optimal bounds on its success probability and advancing understanding of its limitations.

Contribution

It introduces new constructions of hard instances for PPSZ, providing tighter upper bounds on its success probability and improving previous bounds.

Findings

Weak PPSZ has at most (2+ε)/k savings on certain graph instances.

Strong PPSZ has at most O(log(k)/k) savings on linear system instances.

Results are close to known lower bounds, indicating near-optimal hardness constructions.

Abstract

We construct uniquely satisfiable $k$-CNF formulas that are hard for the algorithm PPSZ. 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 $\frac{\pi^2 + o(1)}{6k}$, this is optimal up to the constant factor. In particular, for $k=3$, our upper bound is $2^{0.333\dots n}$, which is fairly 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 from the 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].