A multivariate framework for weighted FPT algorithms

We introduce a multivariate approach for solving weighted parameterized problems. By allowing flexible use of parameters, our approach defines a framework for applying the classic bounded search trees technique. In our model, given an instance of size n of a minimization/maximization problem, and a parameter W≥1, we seek a solution of weight at most/at least W. We demonstrate the usefulness of our approach in solving VERTEX COVER, 3-HITTING SET, EDGE DOMINATING SET and MAX INTERNAL OUT-BRANCHING. While the best known algorithms for these problems admit running times of the form c^Wn^O(1), for some c>1, our framework yields running times of the form c^sn^O(1), where s≤W is the minimum size of a solution of weight at most/at least W. If no such solution exists, s=min⁡{W,m}, where m is the maximum size of a solution. In addition, we analyze the parameter t≤s, the minimum size of a solution.