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.
Categories: cs.DS