Exploiting dominance conditions for computing worst-case time upper bounds in bounded combinatorial optimization problems: application to MIN SET COVERING and MAX CUT1

In the design of branch and bound methods for NP-hard combinatorial optimization problems, dominance conditions have always been applied. In this work we show how the use of dominance conditions within search tree algorithms can lead to non trivial worst-case upper time bounds for the considered algorithms on bounded combinatorial optimization problems. We consider here the MIN 3-SET COVERING problem and the MAX CUT problem in graphs maximum degree three, four, five and six. Combining dominance conditions and intuitive combinatorial arguments, we derive two exact algorithms with worst-case complexity bounded above by p · O(1.4492n) for the former, and p1 · O(1.2920n), p2 · O(1.4142n), p3 · O(1.6430n) and p3 · O(1.6430n), respectively, the latter problems, where p(·) and pi(·), i = 1, . . . , 6, denote some polynomials and n is the number of subsets for MIN 3-SET COVERING and the number of vertices of the input-graph for MAX CUT.