Streaming Algorithms for Budgeted k-Submodular Maximization Problem

Canh V. Pham, Quang C. Vu, Dung K. Ha, Tai T. Nguyen

Abstract: 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 \(\mathbf{s }=(S_1, S_2, \ldots , S_k)\), 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 \(f(\mathbf{s })\) 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 \(\beta =\max _{e\in V, i , j \in [k], i\ne j} \frac{c_i(e)}{c_j(e)}\) and \(\epsilon , \alpha \) are fixed inputs.