The impact of lexicographic parsimony pressure for ORDER/MAJORITY on the run time

While many optimization problems work with a fixed number of decision variables and thus a fixed-length representation of possible solutions, genetic programming (GP) works on variable-length representations. A naturally occurring problem is that of bloat, that is, the unnecessary growth of solution lengths, which may slow down the optimization process. So far, the mathematical runtime analysis could not deal well with bloat and required explicit assumptions limiting bloat. In this paper, we provide the first mathematical runtime analysis of a GP algorithm that does not require any assumptions on the bloat. Previous performance guarantees were only proven conditionally for runs in which no strong bloat occurs. Together with improved analyses for the case with bloat restrictions our results show that such assumptions on the bloat are not necessary and that the algorithm is efficient without explicit bloat control mechanism. More specifically, we analyzed the performance of the (1+1) GP on the two benchmark functions ORDER and MAJORITY. When using lexicographic parsimony pressure as bloat control, we show a tight runtime estimate of O(T_init+nlog⁡n) iterations both for ORDER and MAJORITY. For the case without bloat control, the bounds O(T_initlog⁡T_init+n(log⁡n)³) and Ω(T_init+nlog⁡n) (and Ω(T_initlog⁡T_init) for n=1) hold for MAJORITY.¹