Beyond No Free Lunch: Realistic algorithms for arbitrary problem classes

In this paper we present a simple and general new No Free Lunch-like result that applies to revisiting algorithms searching arbitrary problem sets. We begin by unifying the assumptions of closure under permutation and non-revisiting algorithms. We then propose a new approach to reasoning about search algorithm performance, treating search algorithms as stochastic processes and thereby admitting revisiting; for this approach we need only make a simple assumption that search algorithms are applied for optimisation (i.e. maximisation or minimisation), rather than considering arbitrary performance measures. As a consequence of a proof that non-revisiting enumeration has the best possible expected performance for arbitrary distributions over arbitrary problem sets, this allows us to show that better than enumeration performance by some algorithm on some problem set predicts nothing about performance on a second problem set when nothing is known about the relation between the two. Finally we note that enumeration typically has excellent time and space complexities. Our approach is both simple and powerful; our new No Free Lunch-like result is stronger than the originals both in its predictions and, apart from not applying to arbitrary performance measures but only to those reflecting the typical goal of search, weaker than the originals in its assumptions. It is also directly relevant to research into real-world search algorithms.