Approximation Algorithm for Connected Submodular Function Maximization Problems

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.