Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search

Abstract

Local search is widely used to solve combinatorial optimisation problems and to model biological evolution, but the performance of local search algorithms on different kinds of fitness landscapes is poorly understood. Here we consider how fitness landscapes can be represented using valued constraints, and investigate what the structure of such representations reveals about the complexity of local search.
 First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed. Second, we consider landscapes as equivalent if they allow the same (improving) local search moves; we show that a minimal constraint graph still exists, but is NP-hard to compute.
 We then develop several techniques to bound the length of any sequence of local search moves. We show that such a bound can be obtained from the numerical values of the constraints in the representation, and show how this bound may be tightened by considering equivalent representations. In the binary Boolean case, we prove that a degree 2 or treestructured constraint graph gives a quadratic bound on the number of improving moves made by any local search; hence, any landscape that can be represented by such a model will be tractable for any form of local search.
 Finally, we build two families of examples to show that the conditions in our tractability results are essential. With domain size three, even just a path of binary constraints can model a landscape with an exponentially long sequence of improving moves. With a treewidth-two constraint graph, even with a maximum degree of three, binary Boolean constraints can model a landscape with an exponentially long sequence of improving moves.

The Design of Evolutionary Algorithms: A Computer Science Perspective on the Compatibility of Evolution and Design

“…In trying to understand better the time required for an evolutionary process to find a local peak, Kaznatcheev (2019) introduced a scheme for categorizing fitness landscapes as “easy” or “hard.” He notes that there are families of fitness functions where, from some starting points, following improving paths can trace out exponentially many steps, before reaching a local peak. How common such landscapes are is not yet clear, and it has been shown (Kaznatcheev, Cohen, and Jeavons 2020) that restricting the nature of the fitness function to some simple forms can ensure that all improving paths are of a more practical length (i.e., a polynomial function of the number of variables). However, the mere existence of landscapes with exponentially long paths to the nearest peak serves to counteract the oversimplifying intuition that fitness landscapes all resemble gently rolling hills.…”

The Contentious Compatibility of Evolution and Design: Introduction to the Book Symposium

“…The computer scientist Peter Jeavons (University of Oxford) has led significant work on evolutionary search and theoretical evolutionary biology (Nichol et al 2019; Kaznatcheev, Cohen, and Jeavons 2020). The problem of evolutionary search as encountered in biology is a subset of a larger class of possible searches considered in computer science.…”