Online Set Cover with Happiness Costs

: The Online Set Cover problem (OSC) and its variations are one of the most well-studied optimization problems in operations research and computer science. In OSC, we are given a universe of elements and a collection of subsets of the universe. Each subset is associated with a cost. As elements arrive over time, the algorithm purchases subsets to cover these elements. In each step, an element arrives, and the algorithm needs to ensure that at the end of the step, there is at least one purchased subset that contains the element. The goal is to minimize the total cost of purchased subsets. In this paper, we study a generalization of OSC, in which a request consisting of a number of elements arrives in each step. Each request is associated with a happiness cost. A request is served by either a single subset containing all of its elements or by a number of subsets jointly containing all of its elements. In the latter case, the algorithm needs to pay the happiness cost associated with the request. The goal is to serve all requests upon their arrival while minimizing the total cost of purchased subsets and happiness costs paid. This problem is motivated by intrinsic service-providing scenarios in which clients need not only be served but are to be satisfied with the service. Keeping clients happy by serving them with one service provider rather than many, is represented by happiness costs. We refer to this problem as Online Set Cover With Happiness Costs (OSC-HC) and design the first online algorithm, which is optimal under the competitive analysis framework. The latter is a worst-case analysis framework and the standard to measure online algorithms. It compares, for all instances of the problem, the performance of the online algorithm to that of the optimal offline algorithm that is given all the input sequence at once and is optimal.