text: We study the problem of learning to partition users into groups, where one must learn the compatibilities between the users to achieve optimal groupings. We define four natural objectives that optimize for average and worst case compatibilities and propose new algorithms for adaptively learning optimal groupings. When we do not impose any structure on the compatibilities, we show that the group formation objectives considered are @math hard to solve and we either give approximation guarantees or prove inapproximability results. We then introduce an elegant structure, namely that of , that makes many of these problems polynomial time solvable. We explicitly characterize the optimal groupings under this structure and show that the optimal solutions are related to and partitions, well-studied in the psychology literature. For one of the four objectives, we show @math hardness under the score structure and give a @math approximation algorithm for which no constant approximation was known thus far. Finally, under the score structure, we propose an online low sample complexity PAC algorithm for learning the optimal partition. We demonstrate the efficacy of the proposed algorithm on synthetic and real world datasets.
  @cite_10: This paper concerns the optimal partition of a graph into p connected clusters of vertices, with various constraints on their topology and weight. We consider different objectives, depending on the cost of the trees spanning the clusters. This rich family of problems mainly applies to telecommunication network design, but it can be useful in other fields. We achieve a complete characterization of its computational complexity, previously studied only for special cases: a polynomial algorithm based on a new matroid solves the easy cases; the others are strongly NP-hard by direct reduction from SAT. Finally, we give results on special graphs.
summary: There is one class of graph partitioning problems that come with size restrictions, namely, the Graph Tree Partitioning Problem @cite_10 , where the objective is to partition a graph into equal size subsets, such that the weight of the spanning tree on each subset is either as high as possible (max-min) or as low as possible (min-max), or, the sum of the weights of the spanning trees are either as high (max-sum), or, as low (min-sum) as possible. Though these objectives are closely related to ours, they are not exactly the same problems.