Fully-Dynamic Submodular Cover with Bounded Recourse - Gupta, Anupam; Levin, Roie (Proceedings / annual Symposium on Foundations of Computer Science, 2020)
In submodular covering problems, we are given a monotone, nonnegative submodular function f:2^{\mathcal{N}}→ \mathbb{R}_{+} and wish to find the min-cost set S\subseteq \mathcal{N} such that f(S)=f(\mathcal{N}) . When f is a coverage function, this captures Setcover as a special case. 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 (a.k.a. recourse) is allowed. For concreteness, suppose a nonnegative monotone submodular integer-valued function g_{t} is added or removed from an active set G^{(t)} at each time t . If f^{(t)}=\sum\nolimits_{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. For example, if each g_{t} is the (weighted) rank function of a matroid, we would be dynamically maintaining a low-cost common spanning set for a changing collection of matroids. 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})·\sum\nolimits_{t < T}g_{t}(\mathcal{N})) , where c_{\min}, c_{\min} are the largest/smallest costs of elements in \mathcal{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 sub-modular functions that also have positive mixed third derivatives, we show an optimal recourse bound of O(\sum\nolimits_{t < T}g_{t}(\mathcal{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.