Fully-Dynamic Submodular Cover with Bounded Recourse

TL;DR

This work tackles the fully-dynamic Submodular Cover problem where the objective function f^{(t)} evolves over time as a sum of active submodular components. It introduces a permutation-based dynamic greedy framework and a Tsallis-entropy-inspired potential to bound recourse, achieving near-online competitive guarantees. The results unify and extend prior recourse-bounded dynamic covering, providing optimal recourse for the structured class of 3-increasing functions and improving robustness to weighted costs through Mutual Coverage analysis. The framework applies broadly, including dynamic Set Cover and Hypergraph Vertex Cover, and offers practical, scalable approaches for maintaining coverage under time-varying requirements.

Abstract

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.

Figures (3)

• Figure 1: Illustration of a legal $\gamma$-move. Each rectangle represents the marginal coverage of an element of the permutation. The height of the item that moves must be at least $\gamma$ times the height of anyone it cuts in line.
• Figure 2: Illustration of the proof of \ref{['claim:lvlcost']}.
• Figure 3: Illustration of $\Phi_{1/2}$. Elements are arranged in order of $\pi$.

Theorems & Definitions (50)

• Theorem 1.1: Informal
• Theorem 1.2: Informal
• Definition 1.1: Mutual Coverage
• Theorem 2.1
• Lemma 2.1
• proof
• Corollary 2.1
• proof
• Lemma 2.2
• proof