Approximation Algorithm for Connected Submodular Function Maximization Problems

Wenzheng Xu, He Xue, Jing Li, Weifa Liang, Zichuan Xu, Pan Zhou, Xiaohua Jia, Sajal K. Das

Abstract: In this paper, we study a connected submodular function maximization problem, which arises from many applications including deploying UAV networks to serve users and placing sensors to cover Points of Interest (PoIs). Specifically, given a budget $K$, the problem is to find a subset $S$ with $K$ nodes from a graph $G$ so that a given submodular function $f (S)$ on $S$ is maximized while the induced subgraph $G[S]$ by the nodes in $S$ is connected, where the submodular function $f$ can be used to model many practical application problems, such as the number of users within different service areas of the deployed UAVs in $S$, the sum of data rates of users served by the UAVs, the number of covered PoIs by placed sensors, etc. We then propose a novel $\frac{1-1/e}{2h+2}$ -approximation algorithm for the problem, improving the best approximation ratio $\frac{1-1/e}{2h+3}$ for the problem so far, through estimating a novel upper bound on the problem and designing a smart graph decomposition technique, where $e$ is the base of the natural logarithm, $h$ is a parameter depends on the problem and its typical value is 2. In addition. when $h= 2$, the algorithm approximation ratio is at least $\frac{1-1/e}{5}$ and may be as large as 1 in some special cases when $K$ ≤21, and is no less than $\frac{1-1/e}{6}$ when $K$ ≥ 22, compared with the current best approximation ratio $\frac{1-1/e}{7}(= \frac{1-1/e}{2h+3})$ for the problem. We finally evaluate the algorithm performance in the application of deploying a UAV network. Experimental results demonstrate the number of users within the service area of the deployed UAV network by the proposed algorithm is up to 7.5% larger than those by existing algorithms, and its empirical approximation ratio is between 0.7 and 0.99, which is close to the theoretical maximum value one.