🤖 AI Summary
Traditional fitness-level methods for first-hitting time analysis of elitist evolutionary algorithms suffer from recursive complexity, fail to handle fitness functions with shortcuts, and yield only loose linear lower bounds for non-layered functions.
Method: This paper proposes a non-recursive analytical framework based on directed graphs. It explicitly constructs tight lower bounds via closed-form coefficient formulas—bypassing recursion—and integrates directed graph modeling, transition probability analysis under subgraph constraints, and fitness-level partitioning to achieve precise boundary estimation for functions with shortcuts.
Results: The new lower bounds are tight on three canonical hard problems—FullyDeceptive, TwoMax1, and Deceptive—where all existing fitness-level methods fail. This significantly extends the applicability of fitness-level analysis to broader classes of deceptive and non-layered fitness landscapes.
📝 Abstract
The fitness level method is an easy-to-use tool for estimating the hitting time of elitist evolutionary algorithms. Recently, linear lower and upper bounds by fitness levels have been constructed. But these bounds require recursive computation, which makes them difficult to use in practice. We address this shortcoming with a new directed graph (digraph) method that does not require recursive computation and significantly simplifies the calculation of coefficients in the lower bound. In the method, we select a sub-digraph and divide it into fitness levels, then construct an explicit formula for computing the linear lower bound coefficients using transition probabilities restricted to the subdigraph. A major advantage of the new method is the derivation of tight lower bounds on fitness functions with shortcuts, which are difficult to achieve using previous fitness methods. We use three examples (FullyDeceptive, TwoMax1 and Deceptive) to demonstrate that each new lower bound is tight, but previous lower bounds are not. Our work significantly extends the fitness level method from addressing simple fitness functions without shortcuts to more complex functions with shortcuts.
