Regularized two-stage submodular maximization under streaming

Abstract

In the problem of maximizing regularized two-stage submodular functions in streams, we assemble a family ${\cal F}$ of m functions each of which is submodular and is visited in a streaming style that an element is visited for only once. The aim is to choose a subset S of size at most ℓ from the element stream ${\cal V}$, so as to maximize the average maximum value of these functions restricted on S with a regularized modular term. The problem can be formally casted as ${\max _{S \subseteq V,\left| S \right| \leqslant \ell }}{1 \over m}\sum\nolimits_{i = 1}^m {{{\max }_{T \subseteq S,\left| T \right| \leqslant k}}\left[ {{f_i}\left( T \right) - c\left( T \right)} \right]}$, where $c:{\cal V} \to {\mathbb{R}_ + }$ is a non-negative modular function and ${f_i}:{2^{\cal V}} \to {\mathbb{R}_ + },\forall i \in \left\{ {1, \ldots ,m} \right\}$ is a non-negative monotone non-decreasing submodular function. The well-studied regularized problem of ${\max _{S \subseteq {\cal V},\left| S \right| \leqslant k}}f(S) - c(S)$ is exactly a special case of the above regularized two-stage submodular maximization by setting m = 1 and ℓ = k. Although f(·) − c(·) is submodular, it is potentially negative and non-monotone and admits no constant multiplicative factor approximation. Therefore, we adopt a slightly weaker notion of approximation which constructs S such that f(S) − c(S) ⩾ ρ · f(O) − c(O) holds against optimum solution O for some ρ ∈ (0, 1). Eventually, we devise a streaming algorithm by employing the distorted threshold technique, achieving a weaker approximation ratio with ρ = 0.2996 for the discussed regularized two-stage model.