🤖 AI Summary
This paper studies the dynamic submodular cover problem: maintaining a near-optimal solution under monotone nonnegative submodular functions ( f(t) = sum_{g_t in mathcal{A}_t} g_t ) that evolve over time via element insertions and deletions, while minimizing total recourse (i.e., adjustments to the solution). We propose a novel potential function based on Tsallis entropy and introduce a generalized mutual coverage analysis framework. Our approach achieves the first optimal competitive ratio of ( O(log(f_{max}/f_{min})) ) for general monotone submodular functions. The total recourse is bounded by ( O(log(c_{max}/c_{min}) cdot sum g_t(N)) ), and for submodular functions with nonnegative third-order differences (e.g., set cover), it attains the optimal bound ( O(sum g_t(N)) ). Our method unifies, simplifies, and improves prior results for dynamic SetCover and extends naturally to applications including dynamic set cover and finding common bases in multiple matroids.
📝 Abstract
In submodular covering problems, we are given a monotone, nonnegative submodular function $f:2^{mathcal{N}}
ightarrow mathbb{R}_{+}$ and wish to find the min-cost set $Ssubseteq mathcal{N}$ 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 $f^{(t)}=sum
olimits_{gin G^{(t)}}g$ 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 $O(log(c_{max}/c_{min})cdotsum
olimits_{t < T}g_{t}(mathcal{N}))$, 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 $O(sum
olimits_{t < T}g_{t}(mathcal{N}))$. 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.
