Budget-constrained profit maximization without non-negative objective assumption in social networks

In this paper, we study the budget-constrained profit maximization problem with expensive seed endorsement, a derivation of the well-studied influence maximization and profit maximization in social networks. While existing research requires the non-negativity of the objective profit function, this paper considers real-world scenarios where costs may surpass revenue. Specifically, our problem can be regarded as maximizing the difference between a non-negative submodular function and a non-negative modular function under a knapsack constraint, allowing for negative differences. To tackle this challenge, we propose two algorithms. Firstly, we employ a twin greedy and enumeration technique to design a polynomial-time algorithm with a quarter weak approximation ratio, providing a balance between computational efficiency and solution quality. Then, we incorporate a threshold decreasing technique to enhance the time complexity of the first algorithm, yielding an improved computational efficiency while maintaining a reasonable level of solution accuracy. To our knowledge, this is the first paper to study the profit maximization beyond non-negativity and to propose polynomial-time algorithms with a constant bicriteria approximation ratio.