FullyDynamic Submodular Cover with Bounded Recourse
Abstract
In submodular covering problems, we are given a monotone, nonnegative submodular function $f: 2^N \rightarrow\mathbb{R}_+$ and wish to find the mincost 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 fullydynamic 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 setcoverage functions, so our algorithm matches the known $O(\log n)$competitiveness and $O(1)$ recourse guarantees for fullydynamic 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 arXiv:
 arXiv:2009.00800
 Bibcode:
 2020arXiv200900800G
 Keywords:

 Computer Science  Data Structures and Algorithms