We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J. ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size $n$ which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized $α$-approximation algorithm that runs in $c^k \cdot n^{O(1)}$ time, where $k$ is the solution size, can be used to derive an $α$-approximation randomized algorithm that runs in $d^n \cdot n^{O(1)}$ time, where $d$ is the unique value in $d \in (1,1+\frac{c-1}α)$ such that $\mathcal{D}(\frac{1}α\|\frac{d-1}{c-1})=\frac{\ln c}α$ and $\mathcal{D}(a \|b)$ is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for $α=1$, and is strictly better when $α>1$, for any $c > 1$. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, $3$-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a $1.1$-approximation algorithm for Vertex Cover with running time $1.114^n \cdot n^{O(1)}$, improving upon the previously best known $1.1$-approximation running in time $1.127^n \cdot n^{O(1)}$ by Bourgeois et al. [DAM 2011].

翻译：我们推广了Fomin、Gaspers、Lokshtanov和Saurabh [J. ACM 2019]的单调局部搜索方法，通过建立单调子集最小化问题的参数化近似与指数时间近似算法之间的联系。在一个单调子集最小化问题中，输入隐式地描述了在规模为$n$的全域上的一个非空集合族，该集合族在取超集操作下封闭。任务是在该族中找到一个最小基数的集合。概括地说，我们利用近似单调局部搜索证明：一个运行时间为$c^k \cdot n^{O(1)}$的参数化$α$-近似算法（其中$k$是解的大小）可用于推导出一个运行时间为$d^n \cdot n^{O(1)}$的$α$-近似随机算法，其中$d$是区间$d \in (1,1+\frac{c-1}α)$内满足$\mathcal{D}(\frac{1}α\|\frac{d-1}{c-1})=\frac{\ln c}α$的唯一值，而$\mathcal{D}(a \|b)$是Kullback-Leibler散度。当$α=1$时，该运行时间与Fomin等人的结果一致；对于任意$c > 1$，当$α>1$时，该运行时间严格更优。此外，我们还证明了这一结果可以通过在运行时间中引入一个亚指数乘性因子来进行去随机化。我们通过为顶点覆盖、$3$-命中集、有向反馈顶点集、有向子集反馈顶点集、有向奇环横截集和无向多割问题推导出新的、更快的指数近似算法，展示了近似单调局部搜索的潜力。例如，我们得到了一个运行时间为$1.114^n \cdot n^{O(1)}$的顶点覆盖$1.1$-近似算法，改进了Bourgeois等人[DAM 2011]先前已知的运行时间为$1.127^n \cdot n^{O(1)}$的$1.1$-近似算法。