The Online Submodular Cover Problem

TL;DR

The paper addresses the online submodular cover problem where a time-monotone sequence of monotone submodular functions $f^{(t)}$ on a ground set $N$ must be covered by increasing subsets $S_t$ with $f^{(t)}(S_t)=f^{(t)}(N)$. It introduces an online LP-based approach combined with randomized rounding of the multilinear extension and a rounding-with-alterations step, achieving a competitive ratio of $O(\ln n \; \ln (T \cdot f(N) / f_{\min}))$ in general and a tighter $O(\ln n \; \ln (T \cdot f_{\max} / f_{\min}))$ for $3$-increasing functions; a polynomial-time implementation via a round-or-separate method is also developed. The offline perspective yields an $O(\ln (f(N) / f_{\min}))$-approximation via LP rounding, matching the greedy baseline up to constants, while the online analysis leverages mutual coverage and higher-order monotonicity to tighten guarantees. The results have implications for evolving network monitoring and resource allocation tasks, enabling provably near-optimal incremental coverage as requirements change over time.

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 $O(\ln n \ln (T \cdot f(N) / f_{\text{min}}))$, where $f_{\text{min}}$ 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.

Figures (1)

• Figure 1: Illustration of the charging scheme. The white areas initially represent the marginal values of each element of $\textsc{Opt}\xspace$ conditioned on the elements that come before it in the ordering. The red areas represent the contribution of $G_1$ to the elements of $\textsc{Opt}\xspace$, or the amount by which the marginal value of each element in $\textsc{Opt}\xspace$ changes after conditioning on $G_1$. Likewise, the purple areas represent the contribution of $G_2$ to the elements of $\textsc{Opt}\xspace$.

Theorems & Definitions (32)

• Theorem 1: Main Theorem
• Theorem 2
• Definition : Time-monotone
• Definition : $m$-increasing DBLP:journals/mor/FoldesH05
• Definition : Mutual Coverage
• Lemma 4: Proposition 2 of Wolsey1982
• Theorem 5
• Lemma 6
• proof
• Lemma 7