A multivariate approach for weighted FPT algorithms

Abstract

We introduce a multivariate approach for solving weighted parameterized problems. Building on the flexible use of certain parameters, our approach defines a new general 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 wide applicability of our approach by solving the weighted variants of 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 aWnO(1), for some constant a > 1, our approach yields running times of the form bsnO(1), for some constant b ≤ a, 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. Clearly, s can be substantially smaller than W. Moreover, we give an example for a problem whose polynomialtime solvability crucially relies on our flexible (in lieu of a strict) use of parameters. We further show, among other results, that Weighted VertexCover and Weighted Edge Dominating Set are solvable in times 1.443tnO(1) and 3tnO(1), respectively, where t ≤ s is the minimum size of a solution.