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

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Abstract
A dominance relation D is a binary relation defined on the set of partial problems generated in a branch-and-bound algorithm, such that PiDPj (where Pi and Pj are partial problems) implies that Pj can be excluded from consideration without loss of optimality of the given problem if Pi has already been generated when Pj 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 PiDPj always implies PiD′Pj. 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.