Connected Submodular Maximization (CSM) is a graph problem with important applications to wireless network deployment, path planning, epidemic outbreaks, and cancer genome studies. In CSM, we are given a graph $G$, a non-negative monotone submodular function $f$ on subsets of the vertex set of $G$, and an integer $k$. The goal is to select a tree in $G$, with $k$ edges, whose vertex set maximizes $f$. We also study the more general Directed and Directed Rooted variants of CSM (DCSM and DRCSM respectively). In both variants, $G$ is directed and the solution must be an out-tree in $G$, with $k$ edges, whose vertex set maximizes $f$; DRCSM further specifies a vertex to be the root of the selected out-tree. For CSM, several previous works have proposed polynomial time approximation algorithms; the state-of-the-art polynomial time algorithm achieves a $Ω(\frac{1}{\sqrt{k}})$-approximation. We can also parameterize the approximation factor by the radius of the optimal solution, denoted by $r$; the state-of-the-art polynomial time algorithm achieves a $Ω(\frac{1}{r})$-approximation. In this paper, we improve on the state-of-the-art approximation factor for CSM with respect to $r$ as well as $k$, noting that $r \leq k$. We propose a polynomial time framework that, for (Directed) CSM, achieves a $Ω(\frac{\varepsilon^{3}}{{r}^{\varepsilon}})$-approximation for every constant $\varepsilon \in (0, 1]$. For DRCSM, our framework achieves a $Ω(\frac{δ\varepsilon^{3}}{{r}^{\varepsilon}})$-approximation that violates the size constraint by at most a factor of $1 + δ$ for every $δ\in [\frac{1}{k}, 1]$. A key component of our framework is GreedyRadius, which is an algorithm for DRCSM that takes another algorithm with a bicriteria approximation factor in terms of $k$ and outputs a solution with the same bicriteria approximation factor (up to constants) in terms of $r$.

翻译：连通子模最大化（Connected Submodular Maximization, CSM）是一个图论问题，在无线网络部署、路径规划、疫情爆发及癌症基因组研究中具有重要应用。在CSM中，给定一个图$G$、一个定义在$G$顶点集子集上的非负单调子模函数$f$，以及一个整数$k$。目标是在$G$中选取一棵包含$k$条边的树，使其顶点集最大化$f$。我们还研究了CSM的更一般变体，即有向版本和有向有根版本（分别记为DCSM和DRCSM）。在这两种变体中，$G$是有向图，解必须是$G$中的一棵出树，包含$k$条边，其顶点集最大化$f$；DRCSM进一步指定一个顶点作为所选出树的根。针对CSM，先前已有若干工作提出多项式时间近似算法；当前最优的多项式时间算法实现了$Ω(\frac{1}{\sqrt{k}})$的近似比。我们还可以通过最优解的半径（记为$r$）来参数化近似比；当前最优的多项式时间算法实现了$Ω(\frac{1}{r})$的近似比。本文改进了CSM关于$r$和$k$的当前最优近似比，并注意到$r \leq k$。我们提出一个多项式时间框架，对于（有向）CSM，该框架对任意常数$\varepsilon \in (0, 1]$实现了$Ω(\frac{\varepsilon^{3}}{{r}^{\varepsilon}})$的近似比。对于DRCSM，我们的框架实现了$Ω(\frac{δ\varepsilon^{3}}{{r}^{\varepsilon}})$的近似比，且对每个$δ\in [\frac{1}{k}, 1]$，该近似比在大小约束上的违反因子最多为$1 + δ$。我们框架的一个关键组成部分是GreedyRadius算法，该算法针对DRCSM，接受一个具有关于$k$的双标准近似因子算法，并输出一个关于$r$具有相同双标准近似因子（至多相差常数因子）的解。