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 $\s=(S_1, S_2, \ldots, S_k)$, each element $e \in V$ has a cost $c_i(e)$ to be put into $i$-th set $S_i$, 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 $\frac{1}{4}-ε$ when $f$ is monotone and $\frac{1}{5}-ε$ when $f$ is non-monotone. For the general case, we propose a random streaming algorithm that provides an approximation ratio of $\min\{\fracα{2}, \frac{(1-α)k}{(1+β)k-β} \}-ε$ when $f$ is monotone and $\min\{\fracα{2}, \frac{(1-α)k}{(1+2β)k-2β} \}-ε$ when $f$ is non-monotone in expectation, where $β=\max_{e\in V, i , j \in [k], i\neq j} \frac{c_i(e)}{c_j(e)}$ and $ε, α$ are fixed inputs.