68R01 General
We study the incremental facility location problem, wherein we are given an instance of the uncapacitated facility location problem. We seek an incremental sequence of opening facilities and an incremental sequence of serving customers along with their fixed assignments to facilities open in the partial sequence. Our aim is to have the solution obtained for serving the first l customers in the sequence be competitive with the optimal solution to serve any l customers. We provide an incremental framework that provides an overall competitive factor of 8 and a worst case instance that provides the lower bound of 3. The problem has applications in multi-stage network planning.
Recently, Khuller, Moss and Naor presented a greedy algorithm for the budgeted maximum coverage problem. In this note, we observe that this algorithm also approximates a special case of set-union knapsack problem within a constant factor. In the special case, an element is a member of less than a constant number of subsets. This guarantee naturally extends to densest k-subgraph problem on graphs of bounded degree.