Complexity of evolutionary equilibria in static fitness landscapes

Experiments show that fitness landscapes can have a rich combinatorial structure due to epistasis and yet theory assumes that local peaks can be reached quickly. I introduce a distinction between easy landscapes where local fitness peaks can be found in a moderate number of steps and hard landscapes where finding evolutionary equilibria requires an infeasible amount of time. Hard examples exist even among landscapes with no reciprocal sign epistasis; on these, strong selection weak mutation dynamics cannot find the unique peak in polynomial time. On hard rugged fitness landscapes, no evolutionary dynamics -- even ones that do not follow adaptive paths -- can find a local fitness peak quickly; and the fitness advantage of nearby mutants cannot drop off exponentially fast but has to follow a power-law that long term evolution experiments have associated with unbounded growth in fitness. I present candidates for hard landscapes at scales from singles genes, to microbes, to complex organisms with costly learning (Baldwin effect). Even though hard landscapes are static and finite, local evolutionary equilibrium cannot be assumed.