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

A dominance relation <italic>D</italic> is a binary relation defined on the set of partial problems generated in a branch-and-bound algorithm, such that <italic>P<subscrpt>i</subscrpt>DP<subscrpt>j</subscrpt></italic> (where <italic>P<subscrpt>i</subscrpt></italic> and <italic>P<subscrpt>j</subscrpt></italic> are partial problems) implies that <italic>P<subscrpt>j</subscrpt></italic> can be excluded from consideration without loss of optimality of the given problem if <italic>P<subscrpt>i</subscrpt></italic> has already been generated when <italic>P<subscrpt>j</subscrpt></italic> 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 <italic>D′</italic> is said to be stronger than a dominance relation <italic>D</italic> if <italic>P<subscrpt>i</subscrpt>DP<subscrpt>j</subscrpt></italic> always implies <italic>P<subscrpt>i</subscrpt>D′P<subscrpt>j</subscrpt></italic>. 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.

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