Abstract:
The reconfiguration framework is an emerging framework that concerns finding a step-by-step transformation between two feasible solutions of a problem such that the transformation steps preserve a feasible solution. The complexity (parameterized or otherwise) of the reconfiguration counterpart of several classical NP-hard problems has been studied in the literature over the last few years. In this thesis, we study algorithms revolving around atom reconfiguration. Viewed as a problem on graphs, the atom reconfiguration problem is about finding the most efficient sequence of moves that allows us to go from one atom arrangement to another, atoms being marked vertices. In this context, a move is the movement of an atom through a sequence of edges from some source to some destination such that the selected atom does not collide with any other atom along the way. We define the efficiency of a reconfiguration sequence in terms of number of atom displacements, number of atom extractions/implantations, or a combination of both. We focus on the latter variant. This variant is hard because it is a general case of the atom reconfiguration problem restricted to extraction/implantation minimization, which is known to be NP-complete on general graphs and even when restricted to grid graphs. No matter the variant we are dealing with, our objective function is atom loss, which we aim to minimize. Atom loss is a function of two move operation types applied on atoms, extractions/implantations and displacements, among other variables. We empirically analyze multiple atom reconfiguration solvers, and we prove theorems that aid in orienting our algorithmic work and that may be of independent interest.
