3 Experimental Results

13 seconds. In future work, we intend to try multiple runs, or several populations with inter-population crossovers, and to randomize selection as well as crossover and mutation. It is certainly possible that our local improvement operator, applying the DAGparsimonious algorithm at every stage, is detrimental, or that better local improvement can be done. We intend to try an operator based on a single expansion step of the cost-sharing algorithm. Another possibility is to use DAGparsimonious on some alternate possible topological sorts of the graph (when they exist) and then nd the best of them. Also interesting would be to try crossover of clusters of vertices dependent on the graph structure (clusters), rather than simple 2-point crossover. evaluation of loca improvement operators for genetic algorithms. Presenting experimental run-time results for large problem instances of NP-hard problems is non-trivial, since for some problem instances we do not know the solution. An additional problem is that even if the algorithms run to termination, the large variance of run-time over problem instances makes the comparison noisy, as frequently most of the (averaged over instances) run-time is spent on a single hard instance. A well-known method for comparing such algorithms is counting the number of problem instances (from the same set) that each algorithm solves correctly if allocated a certain time-limit. Since, however, for many problem instances we do not know the correct answer, and since in ab-ductive explanations \the smaller the better", we simply add the number of vertices in the partial results for all problem instances (for a certain run-time limit). The results show that the genetic algorithm fares much better than cost-sharing heuristic search, which was shown to be a good heuristic in [2, 11]. Since the deterministic search algorithm (cost-sharing, here) does not provide a complete set if interrupted, the current best in the algorithm is completed by adding all ancestors of \unexplained" vertices and then running the DAGparsimonious algorithm. The resulting completion, unless the algorithm terminates, is somewhat arbitrary (and non-monotonically decreasing). Thus, to be fair towards the deterministic algorithm, results displayed are for the best result so far. The problem instances were 44 randomly generated directed acyclic graphs with 50 nodes each.