In this paper, we look at the problem of finding the tour of maximum reward
on an undirected graph where the reward is a submodular function, that has a
curvature of κ, of the edges in the tour. This problem is known to be
NP-hard. We analyze two simple algorithms for finding an approximate solution.
Both algorithms require O(∣V∣3) oracle calls to the submodular function. The
approximation factors are shown to be 2+κ1 and
max{3(2+κ)2,2/3(1−κ)}, respectively; so the second
method has better bounds for low values of κ. We also look at how these
algorithms perform for a directed graph and investigate a method to consider
edge costs in addition to rewards. The problem has direct applications in
monitoring an environment using autonomous mobile sensors where the sensing
reward depends on the path taken. We provide simulation results to empirically
evaluate the performance of the algorithms.Comment: Extended version of ACC 2013 submission (including p-system greedy
bound with curvature