On Revealing the Hidden Problem Structure in Real-World and Theoretical Problems Using Walsh Coefficient Influence

Gray-box optimization employs Walsh decomposition to obtain non-linear variable dependencies and utilize them to propose masks of variables that have a joint non-linear influence on fitness value. These masks significantly improve the effectiveness of variation operators. In some problems, all variables are non-linearly dependent, making the aforementioned masks useless. We analyze the features of the real-world instances of such problems and show that many of their dependencies may have noise-like origins. Such noise-caused dependencies are irrelevant to the optimization process and can be ignored. To identify them, we propose extending the use of Walsh decomposition by measuring variable dependency strength that allows the construction of the weighted dynamic Variable Interaction Graph (wdVIG). wdVIGs adjust the dependency strength to mixed individuals. They allow the filtering of irrelevant dependencies and re-enable using dependency-based masks by variation operators. We verify the wdVIG potential on a large benchmark suite. For problems with noise, the wdVIG masks can improve the optimizer's effectiveness. If all dependencies are relevant for the optimization, i.e., the problem is not noised, the influence of wdVIG masks is similar to that of state-of-the-art structures of this kind.

Moving between high-quality optima using multi-satisfiability characteristics in hard-to-solve Max3Sat instances

Gray-box optimization proposes effective and efficient optimizers of general use. To this end, it leverages information about variable dependencies and the subfunction-based problem representation. These approaches were already shown effective by enabling \textit{tunnelling} between local optima even if these moves require the modification of many dependent variables. Tunnelling is useful in solving the maximum satisfiability problem (MaxSat), which can be reformulated to Max3Sat. Since many real-world problems can be brought to solving the MaxSat/Max3Sat instances, it is important to solve them effectively and efficiently. Therefore, we focus on Max3Sat instances for which tunnelling fails to introduce improving moves between locally optimal high-quality solutions and the region of globally optimal solutions. We analyze the features of such instances on the ground of phase transitions. Based on these observations, we propose manipulating clause-satisfiability characteristics that allow connecting high-quality solutions distant in the solution space. We utilize multi-satisfiability characteristics in the optimizer built from typical gray-box mechanisms. The experimental study shows that the proposed optimizer can solve those Max3Sat instances that are out of the grasp of state-of-the-art gray-box optimizers. At the same time, it remains effective for instances that have already been successfully solved by gray-box.