A Lower-Bound Result on the Power of a Genetic Algorithm

This paper presents a lower-bound result on the computational power of a genetic algorithm in the context of combinatorial optimization. We describe a new genetic algorithm, the merged genetic algorithm, and prove that for the class of monotonic functions, the algorithm finds the optimal solution, and does so with an exponential convergence rate. The analysis pertains to the ideal behavior of the algorithm where the main task reduces to showing convergence of probability distributions over the search space of combinatorial structures to the optimal one. We take exponential convergence to be indicative of efficient solvability for the sample-bounded algorithm, although a sampling theory is needed to better relate the limit behavior to actual behavior. The paper concludes with a discussion of some immediate problems that lie ahead.

How Good are Genetic Algorithms at Finding Large Cliques: An Experimental Study

This paper investigates the power of genetic algorithms at solving the MAX-CLIQUE problem. We measure the performance of a standard genetic algorithm on an elementary set of problem instances consisting of embedded cliques in random graphs. We indicate the need for improvement, and introduce a new genetic algorithm, the multi-phase annealed GA, which exhibits superior performance on the same problem set. As we scale up the problem size and test on \hard" benchmark instances, we notice a degraded performance in the algorithm caused by premature convergence to local minima. To alleviate this problem, a sequence of modi cations are implemented ranging from changes in input representation to systematic local search. The most recent version, called union GA, incorporates the features of union cross-over, greedy replacement, and diversity enhancement. It shows a marked speed-up in the number of iterations required to find a given solution, as well as some improvement in the clique size found. We discuss issues related to the SIMD implementation of the genetic algorithms on a Thinking Machines CM-5, which was necessitated by the intrinsically high time complexity (O(n3)) of the serial algorithm for computing one iteration. Our preliminary conclusions are: (1) a genetic algorithm needs to be heavily customized to work "well" for the clique problem; (2) a GA is computationally very expensive, and its use is only recommended if it is known to find larger cliques than other algorithms; (3) although our customization e ort is bringing forth continued improvements, there is no clear evidence, at this time, that a GA will have better success in circumventing local minima.