The Neighbours' Similar Fitness Property for Local Search

For most practical optimisation problems local search outperforms random sampling - despite the "No Free Lunch Theorem". This paper introduces a property of search landscapes termed Neighbours' Similar Fitness (NSF) that underlies the good performance of neighbourhood search in terms of local improvement. Though necessary, NSF is not sufficient to ensure that searching for improvement among the neighbours of a good solution is better than random search. The paper introduces an additional (natural) property which supports a general proof that, for NSF landscapes, neighbourhood search beats random search.

Is perturbation an effective restart strategy?

Premature convergence can be detrimental to the performance of search methods, which is why many search algorithms include restart strategies to deal with it. While it is common to perturb the incumbent solution with diversification steps of various sizes with the hope that the search method will find a new basin of attraction leading to a better local optimum, it is usually not clear how big the perturbation step should be. We introduce a new property of fitness landscapes termed "Neighbours with Similar Fitness" and we demonstrate that the effectiveness of a restart strategy depends on this property.

Steepest ascent can be exponential in bounded treewidth problems

We investigate the complexity of local search based on steepest ascent. We show that even when all variables have domains of size two and the underlying constraint graph of variable interactions has bounded treewidth (in our construction, treewidth 7), there are fitness landscapes for which an exponential number of steps may be required to reach a local optimum. This is an improvement on prior recursive constructions of long steepest ascents, which we prove to need constraint graphs of unbounded treewidth.