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.
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.
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introduce a natural approach to modelling fitness landscapes
consider how fitness landscapes can be represented
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. This allows us to investigate minimal representations (normal forms) and to consider the effects of the structure of the constraint graph on the tractability
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such representations reveals about the complexity of
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normal form
constraint graph
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Next we consider the complexity of local search on fitness landscapes modelled by valued constraints with restricted forms of constraint graph
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
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degree 2 or
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tight
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any form of
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Finally, we
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both
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results
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treewidth two
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