Consider a multi$-$robot coverage problem where a team of N robots is tasked to cover an environment. Each robot i in $[1,,$ N$]$ has a set of candidate actions Xi $,$ from which it can choose one to execute. Suppose S is a set of actions selected by the robots. Denote f$($S$)$ as the joint area covered by the robots, which is assumed to be submodular, considering the coverage overlap. In addition, there is a cost when a robot selects an action. Denote c$($S$)$ as the total cost of the robots, which is assumed to be modular. The objective is to maximize the joint area covered f$($S$)$ while minimizing the total cost c$($S$),$ That is$,$ max S f$($S$)$ c$($S$)$ Is the objective function f$($S$)$ c$($S$)$ submodular? If yes, prove it$.$ If no$,$ give a counter$-$example. $$ Design a greedy algorithm to find S$.$ What is the approximation bound of the greedy algorithm and why? $($Note: you do not have to prove the bound; just explain why.$)$