Local Optima Networks with Escape Edges

Abstract

This paper proposes an alternative definition of edges (escape edges) for the recently introduced network-based model of combinatorial landscapes: Local Optima Networks (LON). The model compresses the information given by the whole search space into a smaller mathematical object that is the graph having as vertices the local optima and as edges the possible weighted transitions between them. The original definition of edges accounted for the notion of transitions between the basins of attraction of local optima. This definition, although informative, produced densely connected networks and required the exhaustive sampling of the basins of attraction. The alternative escape edges proposed here do not require a full computation of the basins. Instead, they account for the chances of escaping a local optima after a controlled mutation (e.g. 1 or 2 bitflips) followed by hill-climbing. A statistical analysis comparing the two LON models for a set of N K landscapes, is presented and discussed. Moreover, a preliminary study is presented, which aims at validating the LON models as a tool for analyzing the dynamics of stochastic local search in combinatorial optimization. K Nv Dedge (%) Lopt all Basin-trans. Esc.D1 Esc.D2 Basin-trans. Esc.D1 Esc.D2

Modelling genetic improvement landscapes with local optima networks

“…Local optima networks (LONs) [9] model the global structure of landscapes as graphs where nodes are local optima and edges represent possible transitions among them with a given search operator. In order to model GI tness landscapes with local optima networks, we adapted the model with escape edges [16]. To construct these networks, we need to de ne their nodes and edges.…”

Local Optima Networks of the Permutation Flow-Shop Problem

“…The connections among optima represent the chances of escaping from a LON N and jumping into another basin after a controlled move [24]. But in a neutral landscape, the partition of solutions into basins of attraction is not sharp: Algorithm 1 is a stochastic operator h and ∀s ∈ S there is a probability p i (s) = P (h(s) ∈ LON N i ).…”

“…This definition, although informative, produced densely connected networks and required exhaustive sampling of the basins of attraction. A second version, escape edges was proposed in [24], which does not require a full computation of the basins. Instead, these edges account for the chances of escaping a local optimum after a controlled mutation (e.g.1 or 2 bit-flips in binary space) followed by hill-climbing.…”

“…Instead, these edges account for the chances of escaping a local optimum after a controlled mutation (e.g.1 or 2 bit-flips in binary space) followed by hill-climbing. This second type of edges has, up to now, only been explored for binary spaces [24]. Also, previous work on networks with both basin and escape edges considered a single move operator on the corresponding search space.…”

Tunnelling Crossover Networks for the Asymmetric TSP

“…This model required a full enumeration of local optima and basins, and was therefore impossible to scale to realistically sized landscapes. An alternative definition of edges was later proposed to account for escape probabilities among optima, that is, probabilities to hop from a local optimum to another after a perturbation (large mutation) followed by local search [18]. Recently, sampling approaches have been developed using escape edges in order to model landscapes of realistic size [6,12,13,11].…”