Abstract
The traditional way of tackling discrete optimization problems is by using local search on suitably defined cost or fitness landscapes. Such approaches are however limited by the slowing down that occurs when the local minima that are a feature of the typically rugged landscapes encountered arrest the progress of the search process. Another way of tackling optimization problems is by the use of heuristic approximations to estimate a global cost minimum. Here, we present a combination of these two approaches by using cover-encoding maps which map processes from a larger search space to subsets of the original search space. The key idea is to construct cover-encoding maps with the help of suitable heuristics that single out near-optimal solutions and result in landscapes on the larger search space that no longer exhibit trapping local minima. We present cover-encoding maps for the problems of the traveling salesman, number partitioning, maximum matching and maximum clique; the practical feasibility of our method is demonstrated by simulations of adaptive walks on the corresponding encoded landscapes which find the global minima for these problems.
Highlights
Fitness landscapes have proved to be a valuable concept in the understanding of adaptation in evolutionary biology and beyond, by visualizing the relationships between genotypes and effective reproductive success (Wright 1932, 1967)
We present cover-encoding maps for the problems of the traveling salesman, number partitioning, maximum matching and maximum clique; the practical feasibility of our method is demonstrated by simulations of adaptive walks on the corresponding encoded landscapes which find the global minima for these problems
Tours of a traveling salesperson problem (TSP) are naturally encoded as permutations of the cities concerned, while spin configurations are encoded as strings over the alphabet {+, −} with each letter referring to a fixed spin variable
Summary
Fitness landscapes have proved to be a valuable concept in the understanding of adaptation in evolutionary biology and beyond, by visualizing the relationships between genotypes and effective reproductive success (Wright 1932, 1967). TSP tours can start at any city so that they are invariant under rotations, while many spin glass models are invariant under simultaneous flipping of all spins This natural or “direct” encoding is often referred to as the phenotype space, see, e.g., (Rothlauf 2006; Neumann and Witt 2010; Rothlauf 2011; Borenstein and Moraglio 2014). What is clear is that, empirically, the introduction of arbitrary redundancy (by means of random Boolean network mapping) does not increase the performance of mutationbased search (Knowles and Watson 2002), suggesting that the inclusion of redundancy should be suitably designed in order to facilitate optimization One such approach was that of Klemm et al (2012), which emphasized the utility of such inhomogeneous genotype–phenotype maps via the idea that low-cost solutions could be enriched and optimization made more efficient in genotype space if the size of the preimage |α−1(x)| of the phenotypes were anti-correlated with the cost function f (x). For such anti-correlations to be imposed, α needs to become explicitly dependent on the cost function
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