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 $\frac{ 1}{ 4}-\epsilon$ 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 $\beta=\max_{ e\in V, i , j \in [k], i\neq j} \frac{ c_i(e)}{ c_j(e)}$ and $\epsilon, \alpha$ are fixed inputs.
Categories: cs.DS cs.GT