Complexity of evolutionary equilibria in static fitness landscapes

TL;DR

This paper demonstrates that finding local fitness peaks in static fitness landscapes can be computationally hard, with some landscapes requiring exponential time for evolution to reach an equilibrium, challenging assumptions of quick adaptation.

Contribution

It proves that the NK model of rugged fitness landscapes is PLS-complete for K >= 2, showing the computational difficulty of reaching local optima in these models.

Findings

NK fitness landscapes are PLS-complete for K >= 2

Adaptive paths can be exponentially long in certain landscapes

Some landscapes lack reciprocal sign epistasis but still have long adaptive paths

Abstract

A fitness landscape is a genetic space -- with two genotypes adjacent if they differ in a single locus -- and a fitness function. Evolutionary dynamics produce a flow on this landscape from lower fitness to higher; reaching equilibrium only if a local fitness peak is found. I use computational complexity to question the common assumption that evolution on static fitness landscapes can quickly reach a local fitness peak. I do this by showing that the popular NK model of rugged fitness landscapes is PLS-complete for K >= 2; the reduction from Weighted 2SAT is a bijection on adaptive walks, so there are NK fitness landscapes where every adaptive path from some vertices is of exponential length. Alternatively -- under the standard complexity theoretic assumption that there are problems in PLS not solvable in polynomial time -- this means that there are no evolutionary dynamics (known, or to be discovered, and not necessarily following adaptive paths) that can converge to a local fitness peak on all NK landscapes with K = 2. Applying results from the analysis of simplex algorithms, I show that there exist single-peaked landscapes with no reciprocal sign epistasis where the expected length of an adaptive path following strong selection weak mutation dynamics is $e^{O(n^{1/3})}$ even though an adaptive path to the optimum of length less than n is available from every vertex. The technical results are written to be accessible to mathematical biologists without a computer science background, and the biological literature is summarized for the convenience of non-biologists with the aim to open a constructive dialogue between the two disciplines.