Bound (Fringe)

In the context of mathematical optimization and computer science, specifically within algorithms like branch and bound, "bound" when referred to as "fringe" describes a concept related to the active nodes or states that are yet to be fully explored.

The fringe, also often called the frontier or open set, represents the collection of currently active nodes in a search tree or exploration graph. These are nodes that have been generated (created) but not yet expanded (had their children or successors explored). They are essentially the boundary between the explored and unexplored portions of the search space.

The concept of "bound" arises in algorithms that use bounds to prune parts of the search space. Branch and bound, for instance, maintains a best-so-far solution. If the bound for a node in the fringe indicates that exploring any of its descendants cannot yield a better solution than the best-so-far, then that node (and its entire subtree) can be safely discarded (pruned). In this sense, the 'bound' is a value or estimate associated with the node in the fringe which guides the search and allows for optimization by avoiding unnecessary exploration.

The efficiency of a branch and bound algorithm is significantly impacted by the quality of the bounds used. Tighter bounds lead to more effective pruning and faster convergence. The structure and organization of the fringe (e.g., using a priority queue based on the bounds) also play a crucial role in the algorithm's performance. Different search strategies (e.g., best-first search, depth-first search) prioritize different nodes in the fringe for expansion based on these bounds and other heuristics.