Patron Allocation for Group Services Under Lower Bound Constraints

Group services are highly important for a variety of computing application domains. In this paper, we study the fundamental problem of allocating a set of service patrons to a set of service groups in an attempt to maximize the total profit gained by the grouping platform. The problem under consideration is unique in that group service is not provided at all unless its lower bound requirement is satisfied. In addition, we allow each service patron to join multiple groups. In this paper, after proving the hardness property of the problem, we focus first on a special case of the problem. To this end, we propose two approaches. One aims at providing a suboptimal solution using a 1/2-approximation algorithm. The other approach turns to seeking an optimal solution using a branch and bound technique. For this purpose, we introduce a theorem that captures a useful property of an optimal allocation. Based on this theorem, we design an efficient branch and bound algorithm to find an optimal solution. We then extend these methods to solve the general problem. Extensive experiments show that our branch and bound algorithm is able to obtain an optimal solution with a small amount of computation time in many different settings.