Average Case Complexity of Branch-and-Bound Algorithms on Random b-ary Trees

Proof. Lets assume that there are two nodes u and v in depth d for which the calculated lower bound is optimal, and u is visited first among all nodes in depth d with optimal lower bound. This implies that the lower bound of v is not smaller than the lower bound of u. A best-first search Branch-and-Bound algorithm only visits nodes with lower bounds not greater than the global minimum. It follows that the lower bound of u must be equal to the global minimum, and at least one leaf node in the sub-tree of u has this minimum value. If the best-first search strategy breaks ties among nodes with equal lower bound by selecting a node with greater depth, all nodes in the subtree of u with lower bound values smaller or equal to the global minimum will be visited before node v, therefore also the leaf node with the global minimum value will be visited before node v. It follows that node v will not be visited, since a leaf node with the global minimum value has already been found. u t