On Self-Avoiding Walks across n-Dimensional Dice and Combinatorial Optimization: An Introduction

Self-avoiding walks (SAWs) were introduced in chemistry to model the reallife behavior of chain-like entities such as solvents and polymers, whose physical volume prohibits multiple occupation of the same spatial point. In mathematics, a SAW lives in the n-dimensional lattice Zn which consists of the points in Rn whose components are integers. In this paper, SAWs are a metaphor for walks across faces of n-dimensional dice, or more formally, a hyperhedron family ) , , ( n b Θ Η . Each face is assigned a label )} ( : { ς ς Θ ; ς represents a unique n-dimensional coordinate string, ) (ς Θ is the value of the function Θ for ς . The walk searches ) (ς Θ for optima by following five simple rules: (1) select a random coordinate and mark it as the ‘initial pivot’; (2) probe all unmarked adjacent coordinates, then select and mark the coordinate with the ’best value’ as the new pivot; (3) continue the walk until either the ’best value’ <= ‘target value’ or the walk is being blocked by adjacent coordinates that are already pivots; (4) if the walk is trapped, restart the walk from a randomly selected ‘new initial pivot’; (5) if needed, manage the memory overflow with a streaminglike buffer of appropriate size. Hard instances from a number of problem domains, including the 2D protein folding problem, with up to (225) * (324) coordinates, have been solved with SAWs in less than 1,000,000 steps – while also exceeding the quality of best known solutions to date.