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.

翻译：在近期工作中，Esmer等人提出了一种简单方法——近似单调局部搜索（Approximate Monotone Local Search），通过现有的参数化精确算法、多项式时间近似算法以及更一般的参数化近似算法，来获得指数时间近似算法。本研究将这些结果推广至加权设置。更严格地说，我们研究在大小为$n$的加权全集上的单调子集最小化问题（例如顶点覆盖、$d$-击中集和反馈顶点集）。我们考虑这样一种模型：算法仅能访问一个子程序，该子程序能在$c^k \cdot n^{O(1)}$时间内找到权重不超过$\alpha \cdot W$（且基数任意）的解，其中$W$是基数至多为$k$的解的最小权重。在无权重设置中，Esmer等人确定了在此模型下能获得运行时间为$d^n \cdot n^{O(1)}$的$\beta$-近似算法的最小值$d$。我们证明，在加权设置下该模型同样保持相同的依赖关系：对于任意固定$\varepsilon>0$，我们可在$O\left((d+\varepsilon)^{n}\right)$时间内获得$\beta$-近似算法，其中$d$与无权重设置中相同。类似地，我们还将$\beta$-近似暴力搜索（在仅提供成员查询预言机的模型中）扩展至加权设置。利用现有的加权问题近似算法和精确参数化算法，我们首次为一系列问题（包括加权顶点覆盖、加权$d$-击中集、加权反馈顶点集和加权多割）获得了优于暴力搜索的指数时间$\beta$-近似算法。