Streaming algorithms for Budgeted $k$-Submodular Maximization problem

Stimulated by practical applications arising from viral marketing. This paper investigates a novel Budgeted $k$-Submodular Maximization problem defined as follows: Given a finite set $V$, a budget $B$ and a $k$-submodular function $f: (k+1)^V \mapsto \mathbb{R}+$, the problem asks to find a solution $\s=(S1, S2, \ldots, Sk)$, each element $e \in V$ has a cost $ci(e)$ to be put into $i$-th set $Si$, with the total cost of $s$ does not exceed $B$ so that $f(\s)$ is maximized. To address this problem, we propose two streaming algorithms that provide approximation guarantees for the problem. In particular, in the case of each element $e$ has the same cost for all $i$-th sets, we propose a deterministic streaming algorithm which provides an approximation ratio of $\frac{1}{4}-\epsilon$ when $f$ is monotone and $\frac{1}{5}-\epsilon$ when $f$ is non-monotone. For the general case, we propose a random streaming algorithm that provides an approximation ratio of $\min{\frac{\alpha}{2}, \frac{(1-\alpha)k}{(1+\beta)k-\beta} }-\epsilon$ when $f$ is monotone and $\min{\frac{\alpha}{2}, \frac{(1-\alpha)k}{(1+2\beta)k-2\beta} }-\epsilon$ when $f$ is non-monotone in expectation, where $\beta=\max{e\in V, i , j \in [k], i\neq j} \frac{ci(e)}{c_j(e)}$ and $\epsilon, \alpha$ are fixed inputs.