Stochastic local search (SLS) is a successful paradigm for solving the satisfiability problem of propositional logic. A recent development in this area involves solving not the original instance, but a modified, yet logically equivalent one. Empirically, this technique was found to be promising as it improves the performance of state-of-the-art SLS solvers. Currently, there is only a shallow understanding of how this modification technique affects the runtimes of SLS solvers. Thus, we model this modification process and conduct an empirical analysis of the hardness of logically equivalent formulas. Our results are twofold. First, if the modification process is treated as a random process, a lognormal distribution perfectly characterizes the hardness; implying that the hardness is long-tailed. This means that the modification technique can be further improved by implementing an additional restart mechanism. Thus, as a second contribution, we theoretically prove that all algorithms exhibiting this long-tail property can be further improved by restarts. Consequently, all SAT solvers employing this modification technique can be enhanced.

Abstract Restarting is a technique frequently employed in randomized algorithms. After some number of computation steps, the state of the algorithm is reinitialized with a new, independent random seed. Luby et al. (Inf. Process. Lett. 47 (4), 173–180, 1993) introduced a universal restart strategy. They showed that their strategy is an optimal universal strategy in the worst case. However, the optimality result has only been shown for discrete processes. In this work, it is shown that their result does not translate into a continuous setting. Furthermore, we show that there are no (asymptotically) optimal strategies in a continuous setting. Nevertheless, we obtain an optimal universal strategy on a restricted class of continuous probability distributions. Furthermore, as a side result, we show that the expected value under restarts for the lognormal distribution tends towards 0. Finally, the results are illustrated using simulations.