# Representing fitness landscapes by valued constraints to understand the complexity of local search

## TL;DR

This paper introduces a novel representation of fitness landscapes using valued constraints, revealing structural properties that determine the complexity of local search and identifying conditions for tractability.

## Contribution

It develops methods to analyze the structure of fitness landscapes via valued constraints, providing bounds on local search complexity and characterizing tractable cases.

## Key findings

- Minimal constraint graphs can be computed efficiently for binary Boolean landscapes.
- NP-hardness of minimal constraint graph computation when landscapes are considered up to local search equivalence.
- Quadratic bounds on local search steps for landscapes with degree 2 or tree-structured constraint graphs.

## 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 tree-structured 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.