The goal of this paper is to understand how exponential-time approximation algorithms can be obtained from existing polynomial-time approximation algorithms, existing parameterized exact algorithms, and existing parameterized approximation algorithms. More formally, we consider a monotone subset minimization problem over a universe of size $n$ (e.g., Vertex Cover or Feedback Vertex Set). We have access to an algorithm that finds an $\alpha$-approximate solution in time $c^k \cdot n^{O(1)}$ if a solution of size $k$ exists (and more generally, an extension algorithm that can approximate in a similar way if a set can be extended to a solution with $k$ further elements). Our goal is to obtain a $d^n \cdot n^{O(1)}$ time $\beta$-approximation algorithm for the problem with $d$ as small as possible. That is, for every fixed $\alpha,c,\beta \geq 1$, we would like to determine the smallest possible $d$ that can be achieved in a model where our problem-specific knowledge is limited to checking the feasibility of a solution and invoking the $\alpha$-approximate extension algorithm. Our results completely resolve this question: (1) For every fixed $\alpha,c,\beta \geq 1$, a simple algorithm (``approximate monotone local search'') achieves the optimum value of $d$. (2) Given $\alpha,c,\beta \geq 1$, we can efficiently compute the optimum $d$ up to any precision $\varepsilon > 0$. Earlier work presented algorithms (but no lower bounds) for the special case $\alpha = \beta = 1$ [Fomin et al., J. ACM 2019] and for the special case $\alpha = \beta > 1$ [Esmer et al., ESA 2022]. Our work generalizes these results and in particular confirms that the earlier algorithms are optimal in these special cases.

翻译：本文旨在理解如何从现有的多项式时间近似算法、现有的参数化精确算法以及现有的参数化近似算法中获得指数时间近似算法。更形式化地，我们考虑在一个规模为$n$的论域上的单调子集最小化问题（例如，顶点覆盖或反馈顶点集）。我们可调用一个算法，该算法在存在规模为$k$的解时，能在时间$c^k \cdot n^{O(1)}$内找到一个$\alpha$-近似解（更一般地，当某个集合可扩展为包含$k$个额外元素的解时，我们有一个能以类似方式进行近似的扩展算法）。我们的目标是为此问题获得一个运行时间为$d^n \cdot n^{O(1)}$的$\beta$-近似算法，并尽可能使$d$最小。即，对于每个固定的$\alpha, c, \beta \geq 1$，我们希望在问题特定知识仅限于检查解的可行性以及调用$\alpha$-近似扩展算法的模型下，确定可达到的最小可能的$d$值。我们的结果彻底解决了这一问题：(1) 对于每个固定的$\alpha, c, \beta \geq 1$，一个简单算法（“近似单调局部搜索”）达到了$d$的最优值。(2) 给定$\alpha, c, \beta \geq 1$，我们可在任意精度$\varepsilon > 0$下有效计算出最优的$d$。早期的研究给出了针对特殊情况$\alpha = \beta = 1$的算法（但无下界）[Fomin等, J. ACM 2019]以及特殊情况$\alpha = \beta > 1$的算法 [Esmer等, ESA 2022]。我们的工作推广了这些结果，并特别证实了这些特殊情况下的早期算法是最优的。