Embedded Landscapes

Abstract

In this paper we introduce embedded landscapes as an extension of NK landscapes and MAXSAT problems. This extension is valid for problems where the representation can be expressed as a simple sum of subfunctions over subsets of the representation domain. This encompasses many additive constraint problems and problems expressed as the interaction of subcomponents, where the critical features of the subcomponents are represented by subsets of bits in the domain. We show that embedded landscapes of fixed maximum epistasis K are exponentially sparse in epistatic space with respect to all possible functions. We show we can compute many important statistical features of these functions in polynomial time including all the epistatic interactions and the statistical moments of hyperplanes about the function mean and hyperplane mean. We also show that embedded landscapes of even small fixed K can be NP-complete. We can conclude that knowing the epistasis and many of the hyperplane statistics is not enough to solve the exponentially difficult part of these general problems and that the difficulty of the problem lies not in the epistasis itself but in the interaction of the epistatic parts.

Efficient Linkage Discovery by Limited Probing

“…The function f : B→ R has k-bounded epistasis if it can be written as the sum of subfunctions each of whose support is a set of at most k bits. It has been shown, perhaps most recently in [Hec02]: …”

Extended probe method for linkage discovery over high-cardinality alphabets

“…Kargupta et al demonstrated that for an epistatically bounded function, i.e. the size of the epistatic subsets is bounded, all the Walsh coefficients could be computed in a polynomial number of function evaluations [8].In 2002, Heckendorn introduced embedded landscapes as extension of NK landscapes and MAXSAT problems [4] and gave theoretical analysis about the relationship between Walsh coefficients and epistases. In 2004, Heckendorn and Wright further built the mathematical foundations for predicting epistasis of binary functions [5].…”

“…Original Walsh function [5,8,4] can not be applied to analyze the high-cardinality domain, so we have to extend it to discrete Fourier function. Some useful properties of Fourier functions are given.…”

Detecting the epistatic structure of generalized embedded landscape

“…The process of discovering the epistatically interacting subsets of parameters is called epistatic discovery or linkage learning [4][5][6][7][8][9][10][11]. Although it is no guarantee of a polynomial time solution to a black box optimization problem [12], it has been realized that competent and scalable performance is best obtained when knowledge of the epistatic structure is used in designing the problem representation and operators or is identified during the optimization [1].…”

“…Algorithms include the messy GA [4], the gene expression messy GA [5], linkage learning GA [6,15], estimation of distribution algorithms (EDAs) [7,8,[16][17][18][19] and linkage detection algorithms [9][10][11]20]. Some of these methods, e.g.…”