Generating Easy and Hard Problems using the Proximate Optimality Principle

We present an approach to generating problems of variable difficulty based on the well-known Proximate Optimality Principle (POP), often paraphrased as "similar solutions have similar fitness". We explore definitions of this concept in terms of metrics in objective space and in representation space and define POP in terms of coherence of these metrics. We hypothesise that algorithms will perform well when the neighbourhoods they explore in representation space are coherent with the natural metric induced by fitness on objective space. We develop an explicit method of problem generation which creates bit string problems where the natural fitness metric is coherent or anti-coherent with Hamming neighbourhoods. We conduct experiments to show that coherent problems are easy whereas anti-coherent problems are hard for local hill climbers using the Hamming neighbourhoods.