Fully Polynomial Time Approximation Scheme for the Pagination Problem with hierarchical structure of tiles

The pagination problem is described as follows. We have a set of symbols, a collection of subsets over these symbols (we call these subsets the tiles) and an integer capacity C. The objective is to find the minimal number of pages (a type of container) on which we can schedule all the tiles while following two fundamental rules. We cannot assign more than C symbols on each page and a tile cannot be broken into several pieces, all of its symbols must be assigned to at least one of the pages. The dierence from the Bin Packing Problem is that tiles can merge. If two tiles share a subset of symbols and if they are assigned to the same page, this subset will be assign only once to the page (and not several times). In this paper, as this problem is NP-complete, we are going to look at a particular case of the dual problem, where we have exactly two pages for which the capacity must be minimized. We are going to present a fully polynomial time approximation scheme (FPTAS) to solve it. This approximation scheme is based on the simplification of a dynamic programming algorithm and it has a strongly polynomial time complexity. The conducted numerical experiments show its practical eectiveness.