## 1 Introduction  
Cost functions appear in many fields of science, economics, and engineering, wherever issues concerning the optimization, typically the minimization or maximization, of some (scalar) quantity--the so-called cost--are raised [1]. Classical examples are the cost of a business strategy, the fitness of biological species or individuals, the objective function value of a trajectory in an optimal control problem, or the potential energy of an arrangement of atoms in space. Given a state space of the system of interest, the cost function assigns a real number to each state of the system; such a state is often called a microstate, configuration, or conformation (in the case of chemical systems), or a legitimate solution to a combinatorial optimization problem. Here, the state space can be a discrete set of states, like the nodes of a graph (often called a metagraph in the context of cost function analysis [2]), or exhibit a continuous structure, like a subspace of \(\mathbb{R}^{n}\). Together with the neighborhood structure of the state space--in the case of a discrete state space, the connectivity of the metagraph--the cost function can be represented by a so-called cost function landscape, which usually contains a multitude of local minima that are separated by a complex barrier structure. In this context, one should note that the choice of connectivity or neighborhood is essentially left to the researcher him/herself; since the choice of the neighborhood implicitly defines the dynamics of the system in most cases, one often speaks of the neighborhood in terms of the moveclass of the system--a terminology derived from the description of the exploration algorithms that are designed to study the landscape. Often, the number of local minima grows exponentially or even factorially with some parameter that represents the size of a system, such as the number of cities in a travelling salesman problem [3; 4], the number of spins in a magnetic material, or the number of atoms in a cluster or molecule [5]. For more general aspects of cost function, fitness, or energy landscapes, we refer to the literature [1]. In the case of a chemical or physical system, one frequently encounters the (potential) energy of the system as the quantity to be studied or minimized, and thus one speaks of the energy landscape of, e.g., the molecule of interest. Furthermore, in the case of such systems, the landscape is of much greater importance than just as a tool to find the lowest energy, since the laws of physics that describe the evolution of the system are embedded in the local shape of the potential energy surface (PES): the gradient of the potential energy yields the forces acting on the atoms, and thus prescribes the subsequent trajectory of, e.g., the atoms belonging to a molecule, such as the vibrations or translational motion of the molecule. Thus, the study of the energy landscape of physical and chemical systems usually goes far beyond the obvious task of identifying the global minimum; for example, all low-energy local minima surrounded by high enough barriers on the landscape correspond to (meta)stable configurations of, e.g., the molecule, which can be observed or synthesized, in principle, or transformed into each other. Therefore, determining all low-energy minima of a chemical system ranks as a goal of similar importance as finding the global minimum [1]. Furthermore, after the initial step of finding not only the global minimum but also as many side-minima as possible, one would study the energy barriers and entropy barriers of the system separating these minima. This requires the design and implementation of new classes of algorithms that, e.g., identify saddle points between local minima [6; 7], or measure the entropic barriers between basins for portions of the landscape restricted to lie below energy lids, such as the threshold algorithm [8].