On finding a minimum dominating set in a tournament

Abstract

The problem of finding a minimum dominating set in a tournament can be solved in n O(log n ) time. It is shown that if this problem has a polynomial-time algorithm, then for every constant C, there is also a polynomial-time algorithm for the satisfiability problem of boolean formulas in conjunctive normal form with m clauses and C log 2 m variables. On the other hand, the problem can be reduced in polynomial time to a general satisfiability problem of length L with O(log 2 L) variables. Another relation between the satisfiability problem and the minimum dominating set in a tournament says that the former can be solved in 2 O(√ v) n K time (where v is the number of variables, n is the length of the formula, and K is a constant) if and only if the latter has a polynomial-time algorithm.