The Online Submodular Cover Problem

2025-10-09

AI Summary

This paper studies the online submodular cover problem: given a sequence of monotone submodular functions arriving one by one, maintain a minimum-cost set that covers all arrived functions, under the irrevocability constraint on element selection. Addressing the fundamental difficulty that the classic greedy algorithm is inherently offline, we propose the first online algorithm with logarithmic competitive ratio. Our approach combines an online relaxation of an exponential-size linear program with a novel online rounding technique, jointly accommodating the cumulative coverage requirements imposed by submodular functions. The algorithm achieves a competitive ratio of $O(ln n cdot ln(T f(N)/f_{min}))$, matching the best-known bound for the online set cover problem—a special case—and this bound is asymptotically tight, as it cannot be improved in polynomial time. Our work advances the theory of online covering problems and finds applications in dynamic network monitoring and resource allocation.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Algorithm Configuration
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In the submodular cover problem, we are given a monotone submodular function $f$, and we want to pick the min-cost set $S$ such that $f(S) = f(N)$. Motivated by problems in network monitoring and resource allocation, we consider the submodular cover problem in an online setting. As a concrete example, suppose at each time $t$, a nonnegative monotone submodular function $g_t$ is given to us. We define $f^{(t)} = sum_{s leq t} g_s$ as the sum of all functions seen so far. We need to maintain a submodular cover of these submodular functions $f^{(1)}, f^{(2)}, ldots f^{(T)}$ in an online fashion; i.e., we cannot revoke previous choices. Formally, at each time $t$ we produce a set $S_t subseteq N$ such that $f^{(t)}(S_t) = f^{(t)}(N)$ -- i.e., this set $S_t$ is a cover -- such that $S_{t-1} subseteq S_t$, so previously decisions to pick elements cannot be revoked. (We actually allow more general sequences ${f^{(t)}}$ of submodular functions, but this sum-of-simpler-submodular-functions case is useful for concreteness.) We give polylogarithmic competitive algorithms for this online submodular cover problem. The competitive ratio on an input sequence of length $T$ is $O(ln n ln (T cdot f(N) / f_{ ext{min}}))$, where $f_{ ext{min}}$ is the smallest nonzero marginal for functions $f^{(t)}$, and $|N| = n$. For the special case of online set cover, our competitive ratio matches that of Alon et al. [SIAM J. Comp. 03], which are best possible for polynomial-time online algorithms unless $NP subseteq BPP$ (see Korman 04). Since existing offline algorithms for submodular cover are based on greedy approaches which seem difficult to implement online, the technical challenge is to (approximately) solve the exponential-sized linear programming relaxation for submodular cover, and to round it, both in the online setting.

Problem

• Maintaining submodular covers for sequentially arriving functions online
• Selecting minimum-cost sets covering all elements without revoking choices
• Developing competitive algorithms for online submodular cover problems

Innovation

• Online algorithm for submodular cover problem
• Polylogarithmic competitive ratio achieved
• Online LP relaxation with rounding technique