Sinking an Algorithmic Isthmus: (1 + ε)-Approximate Min-Sum Subset Convolution

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 \[ (f \star g)(S) = \min_{T \subseteq S}\:\big(f(T) + g(S \setminus T)\big), \] is a fundamental tool in parameterized algorithms. However, since its na\"ive $O(3^n)$-time evaluation is also the fastest known, it has been used only in settings where the input functions have a bounded integer range $\{-M, \ldots, M\}$. In this case, the running time becomes $\tilde O(2^n M)$ by resorting to fast subset convolution in the sum-product ring. This is disadvantageous due to the dependence on $M$, limiting its practicality. In this light, we study whether the problem admits an $(1 + \varepsilon)$-approximation scheme in time independent of $M$. Our main result is the first $\tilde O(2^\frac{3n}{2} / \sqrt{\varepsilon})$-time algorithm for the $(1 + \varepsilon)$-approximate min-sum subset convolution. To show its applicability, we present $(1 + \varepsilon)$-approximation schemes in the same exponential time bound for several NP-hard problems using this convolution, such as the minimum-cost $k$-coloring problem -- in time $\tilde O(2^\frac{3n}{2} / \sqrt{\varepsilon})$, and the prize-collecting Steiner tree problem -- in time $\tilde O(2^\frac{3s^+}{2} / \sqrt{\varepsilon})$, where $n$ is the number of vertices and $s^+$ is the number of proper potential terminals. We also discuss two other applications in computational biology. Our algorithms lie at the intersection of two lines of research that have been considered separately: $\textit{sequence}$ and $\textit{subset}$ convolutions in semi-rings. In particular, we extend the recent framework of Bringmann, K\"unnemann, and W\k{e}grzycki [STOC 2019] to the context of subset convolutions.