Approximate Monotone Local Search for Weighted Problems

In a recent work, Esmer et al. describe a simple method - Approximate Monotone Local Search - to obtain exponential approximation algorithms from existing parameterized exact algorithms, polynomial-time approximation algorithms and, more generally, parameterized approximation algorithms. In this work, we generalize those results to the weighted setting. More formally, we consider monotone subset minimization problems over a weighted universe of size $n$ (e.g., Vertex Cover, $d$-Hitting Set and Feedback Vertex Set). We consider a model where the algorithm is only given access to a subroutine that finds a solution of weight at most $\alpha \cdot W$ (and of arbitrary cardinality) in time $c^k \cdot n^{O(1)}$ where $W$ is the minimum weight of a solution of cardinality at most $k$. In the unweighted setting, Esmer et al. determine the smallest value $d$ for which a $\beta$-approximation algorithm running in time $d^n \cdot n^{O(1)}$ can be obtained in this model. We show that the same dependencies also hold in a weighted setting in this model: for every fixed $\varepsilon>0$ we obtain a $\beta$-approximation algorithm running in time $O\left((d+\varepsilon)^{n}\right)$, for the same $d$ as in the unweighted setting. Similarly, we also extend a $\beta$-approximate brute-force search (in a model which only provides access to a membership oracle) to the weighted setting. Using existing approximation algorithms and exact parameterized algorithms for weighted problems, we obtain the first exponential-time $\beta$-approximation algorithms that are better than brute force for a variety of problems including Weighted Vertex Cover, Weighted $d$-Hitting Set, Weighted Feedback Vertex Set and Weighted Multicut.