  In submodular covering problems, we are given a monotone, nonnegative
submodular function $f: 2^N \rightarrow\mathbb{R}_+$ and wish to find the
min-cost set $S\subseteq N$ such that $f(S)=f(N)$. This captures SetCover when
$f$ is a coverage function. We introduce a general framework for solving such
problems in a fully-dynamic setting where the function $f$ changes over time,
and only a bounded number of updates to the solution (recourse) is allowed. For
concreteness, suppose a nonnegative monotone submodular function $g_t$ is added
or removed from an active set $G^{(t)}$ at each time $t$. If
$f^{(t)}=\sum_{g\in G^{(t)}} g$ is the sum of all active functions, we wish to
maintain a competitive solution to SubmodularCover for $f^{(t)}$ as this active
set changes, and with low recourse.
  We give an algorithm that maintains an $O(\log(f_{max}/f_{min}))$-competitive
solution, where $f_{max}, f_{min}$ are the largest/smallest marginals of
$f^{(t)}$. The algorithm guarantees a total recourse of $O(\log(c_{max}/
c_{min})\cdot\sum_{t\leq T}g_t(N))$, where $c_{max},c_{min}$ are the
largest/smallest costs of elements in $N$. This competitive ratio is best
possible even in the offline setting, and the recourse bound is optimal up to
the logarithmic factor. For monotone submodular functions that also have
positive mixed third derivatives, we show an optimal recourse bound of
$O(\sum_{t\leq T}g_t(N))$. This structured class includes set-coverage
functions, so our algorithm matches the known $O(\log n)$-competitiveness and
$O(1)$ recourse guarantees for fully-dynamic SetCover. Our work simultaneously
simplifies and unifies previous results, as well as generalizes to a
significantly larger class of covering problems. Our key technique is a new
potential function inspired by Tsallis entropy. We also extensively use the
idea of Mutual Coverage, which generalizes the classic notion of mutual
information.
