Note—On the Use of Fictitious Bounds in Tree Search Algorithms

One of the strategies used by many tree search algorithms is to follow down one path in the tree until either a feasible solution is found or else fathoming occurs. For a minimization problem, the lower bounds calculated at various tree nodes tend to be well below the optimal value of the objective function, let alone the values of the successively improving upper bounds. As a result fathoming usually occurs only deep in the tree, and consequently, the search becomes rather lengthy. The purpose of this note is to describe a procedure for using and updating fictitious upper bounds in a systematic way so that optimal and suboptimal solutions can be obtained with a smaller computational effort. The procedure is illustrated by two examples: the traveling salesman problem and the quadratic assignment problem.