The Power of Dominance Relations in Branch-and-Bound Algorithms

A dominance relation D is a binary relation defined on the set of partial problems generated in a branch-and-bound algorithm, such that P i DP j (where P i and P j are partial problems) implies that P j can be excluded from consideration without loss of optimality of the given problem if P i has already been generated when P j is selected for the test. The branch-and-bound computation is usually enhanced by adding the test based on a dominance relation. A dominance relation D′ is said to be stronger than a dominance relation D if P i DP j always implies P i D′P j . Although it seems obvious that a stronger dominance relation makes the resulting algorithm more efficient, counterexamples can easily be constructed. In this paper, however, four classes of branch-and-bound algorithms are found in which a stronger dominance relation always gives a more efficient algorithm. This indicates that the monotonicity property of dominance relations would be observed in a rather wide class of branch-and-bound algorithms, thus encouraging the designer of a branch-and-bound algorithm to find the strongest possible dominance relation.