A EXPERIMENT CONFIGURATIONS Environment Modifications. For the n-Chain environment, the episode length was reduced from 1000 to only 200, what increases the problem difficulty. For the CartPole environment, the input parameters have been re-scaled to the interval [0,1] (using reasonable lower and upper bounds as determined in preliminary runs) since the utilized XCS implementation expects normalized inputs. The reward scheme was slightly modified to deliver +1 every time the pole keeps balanced by the cart and 0 if it has fallen. This provides a clearer reward signal and facilitates quicker learning for each of the compared algorithms. Regarding the MountainCar environment, we again changed the episode length from 200 to 500, since both XCS variants encountered problems in randomly stumbling into the goal state within only 200 steps. Further, the reward scheme was made binary, i.e., instead of paying-1 for every step (includes reaching the terminal/goal state), we chose to apply a 0/1000 scheme. The latter only pays a reward of 1000 when reaching the goal, otherwise 0. This modification essentially increases the problem difficulty, since it renders the reward signal sparse.
The minimum set cover problem (MSCP) is one of the first NP-hard optimization problems discovered. Theoretically it has a bad worst case approximation ratio. As the MSCP turns out to appear in several real world problems, various approaches exist where evolutionary algorithms and metaheuristics are utilized in order to achieve good average case results. This work is intended to revisit and compare current results regarding the application of metaheuristics for the MSCP. Therefore, a recapitulation of the MSCP and its classification into the class of NP-hard optimization problems are provided first. After an overview of notable approximation methods, the focus is shifted towards a brief review of existing metaheuristics which were adapted for the MSCP. In order to allow for a targeted comparison of the existing algorithms, the theoretical worst case complexities in terms of the big O-notation are derived first. This is followed by an empirical study where the identified metaheuristics are examined. Here we use Steiner triple systems, Beasley's OR library, and introduce a new class of instances. Several of the considered approaches achieve close to optimal results. However, our analysis reveals significant differences in terms of runtime and shows that some approaches may even have exponential runtime.
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