A Faster Algorithm for Dominating Set Analyzed by the Potential Method

Measure and Conquer is a recently developed technique to analyze worst-case complexity of backtracking algorithms. The traditional measure and conquer analysis concentrates on one branching at once by using only small number of variables. In this paper, we extend the measure and conquer analysis and introduce a new analyzing technique named "potential method" to deal with consecutive branchings together. In potential method, the optimization problem becomes sparse; therefore, we can use large number of variables. We applied this technique to the minimum dominating set problem and obtained the current fastest algorithm that runs in O(1.4864n) time and polynomial space. We also combined this algorithm with a precalculation by dynamic programming and obtained O(1.4689n) time and space algorithm. These results show the power of the potential method and possibilities of future applications to other problems.