Given functions $f$ and $g$ defined on the subset lattice of order $n$, their min-sum subset convolution, defined for all $S \subseteq [n]$ as \[ (f \star g)(S) = \min_{T \subseteq S}\:\big(f(T) + g(S \setminus T)\big), \] lies at the heart of several NP-hard optimization problems, such as minimum-cost $k$-coloring, the prize-collecting Steiner tree, and many others in computational biology. Despite its importance, its na\"ive $O(3^n)$-time evaluation remains the fastest known, the other alternative being an $\tilde O(2^n M)$-time algorithm for instances where the input functions have a bounded integer range $\{-M, \ldots, M\}$. We study for the first time the $(1 + \varepsilon)$-approximate min-sum subset convolution and present both a weakly- and strongly-polynomial approximation algorithm, running in time $\tilde O(2^n \log M / \varepsilon)$ and $\tilde O(2^\frac{3n}{2} / \sqrt{\varepsilon})$, respectively. To demonstrate the applicability of our work, we present the first exponential-time $(1 + \varepsilon)$-approximation schemes for the above optimization problems. Our algorithms lie at the intersection of two lines of research that have been so far considered separately: $\textit{sequence}$ and $\textit{subset}$ convolutions in semi-rings. We also extend the recent framework of Bringmann, K\"unnemann, and W\k{e}grzycki [STOC 2019] to the context of subset convolutions.

翻译：给定定义在$n$阶子集格上的函数$f$和$g$，其最小和子集卷积（对所有$S \subseteq [n]$定义为\[ (f \star g)(S) = \min_{T \subseteq S}\:\big(f(T) + g(S \setminus T)\big) \]）是若干NP难优化问题的核心，例如最小代价$k$-着色、奖励收集斯坦纳树以及计算生物学中的诸多问题。尽管其重要性不言而喻，但最朴素$O(3^n)$时间的计算仍为已知最快方法，另一种替代方案是在输入函数具有有界整数范围$\{-M, \ldots, M\}$的情况下，采用$\tilde O(2^n M)$时间的算法。我们首次研究$(1 + \varepsilon)$-近似最小和子集卷积，并分别提出弱多项式与强多项式近似算法，运行时间分别为$\tilde O(2^n \log M / \varepsilon)$和$\tilde O(2^\frac{3n}{2} / \sqrt{\varepsilon})$。为展示工作的适用性，我们针对上述优化问题首次提出指数时间$(1 + \varepsilon)$-近似方案。我们的算法处于两条迄今独立研究脉络的交汇处：半环上的序列卷积与子集卷积。同时，我们还扩展了Bringmann、Künnemann和Węgrzycki [STOC 2019] 的最新框架至子集卷积语境。