Judicious partitions and related problems

Many classical partitioning problems in combinatorics ask for a single quantity to be maximized or minimized over a set of partitions of a combinatorial object. For instance, Max Cut asks for the largest bipartite subgraph of a graphG, while Min Bisection asks for the minimum size of a cut into two equal pieces. In judicious partitioning problems, we seek to maximize or minimize a number of quantities simultaneously. For instance, given a graph G with m edges, we can ask for the smallest f(m) such that G must have a bipartition in which each vertex class contains at most f(m) edges. In this survey, we discuss recent extremal results on a variety of questions concerning judicious partitions, and related problems such as Max Cut.