Fully-Dynamic Submodular Cover with Bounded Recourse

Anupam Gupta, Roie Levin

Abstract: In submodular covering problems, we are given a monotone, nonnegative submodular function f:2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> → R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sub> and wish to find the min-cost set S ⊆ N such that f(S)=f(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 gt is added or removed from an active set G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(t)</sup> at each time t. If f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(t)</sup> =Σ(g∈G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(t)</sup> g) is the sum of all active functions, we wish to maintain a competitive solution to Submodularcover for f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(t)</sup> as this active set changes, and with low recourse. For example, if each gt 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> /f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> )) - competitive solution, where f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> , f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> are the largest/smallest marginals of f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(t)</sup> . The algorithm guarantees a total recourse of O(log(c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> /c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> )·Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t≤Tgt</sub> (N)), where c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> , c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> 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 sub-modular functions that also have positive mixed third derivatives, we show an optimal recourse bound of O(Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t≤Tgt</sub> (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.