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 introduce a natural approach to modelling fitness landscapes using valued constraints. This allows us to investigate minimal representations (normal forms) and to consider the effects of the structure of the constraint graph on the tractability 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 normal form still exists, but is NP-hard to compute. Next we consider the complexity of local search on fitness landscapes modelled by valued constraints with restricted forms of constraint graph. In the binary Boolean case, we prove that a tree-structured constraint graph gives a tight 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 local search. We build two families of examples to show that both the conditions in our tractability result 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.

Model genotype–phenotype mappings and the algorithmic structure of evolution

“…De Visser & Krug [28] (also Orr [70]) demonstrated that accessible evolutionary trajectories in empirical landscapes are often short. This result is in contrast to the theoretical result showing that evolutionary trajectories in fitness landscapes can have length exponential in the number of loci [99]. This finding suggests that the GP-mapping induces a specific structure of fitness landscapes in which high fitness is accessible through relatively few mutations.…”

Evolution is exponentially more powerful with frequency-dependent selection

“…Or more formally, if for every ( D n , f n,s ) with probability 1 − δ the population reaches a genotype y with in time polynomial in n , | s |, ln 1 /δ , and 1 /ϵ . Note that this is different from Kaznatcheev [2] and Kaznatcheev, Cohen, and Jeavons [3]’s focus on local peaks.…”

“…The formal approach of algorithmic Darwinism allows us to view evolution as an algorithm. Since evolution is an algorithm, it is subject to all the same information-and computation-theoretic constraints that limit all algorithms [1][2][3]. This allows us to identify families of fitness landscapes or environments that cannot be adapted-to.…”

“…Evolution is an algorithm. As such, evolution is subject to the same constraints of computational complexity as all algorithms [1][2][3]. But evolution is not any algorithm.…”

Steepest ascent can be exponential in bounded treewidth problems

“…We first show in the next section that these examples cannot be expressed by soft constraints whose constraint graph has bounded treewidth. In related work, Kaznatcheev, Cohen, and Jeavons [5] have studied the complementary problem of guaranteeing short paths to a local optimum for local search which performs an arbitrary improving flip at each step rather than a steepest ascent. In the process, they have described some bounded treewidth examples where some exponentially long improving paths exist but these are not the paths followed by steepest ascent.…”