Design by Measure and Conquer: Exact algorithms for dominating set

The measure and conquer approach has proven to be a powerful tool to analyse exact algorithms for combinatorial problems, like Dominating Set and Independent Set. In this paper, we propose to use measure and conquer also as a tool in the design of algorithms. In an iterative process, we obtain a series of branch and reduce algorithms. A mathematical analysis of an algorithm in the series with measure and conquer results in a quasiconvex programming problem. The solution by computer to this problem not only gives a bound on the running time, but can also give a new reduction rule, thus giving a new, possibly faster algorithm. This makes design by measure and conquer a form of computer aided algorithm design. We apply the methodology to a set cover modelling of the Dominating Set problem and obtain the currently fastest known exact algorithm for Dominating Set: an algorithm that uses O(1.4969n) time and only polynomial space, while the previous fastest algorithm uses exponential space.