Hypergraph Partitioning and Clustering

A hypergraph is a generalization of a graph wherein edges can connect more than two vertices and are called hyperedges. Just as graphs naturally represent many kinds of information in mathematical and computer science problems, hypergraphs also arise naturally in important practical problems, including circuit layout, Boolean SATisfiability, numerical linear algebra, etc. Given a hypergraph H , k-way partitioning of H assigns vertices of H to k disjoint nonempty partitions. The k-way partitioning problem seeks to minimize a given cost function of such an assignment. A standard cost function is net cut, which is the number of hyperedges that span more than one partition, or, more generally, the sum of weights of such edges. Constraints are typically imposed on the solution, and make the problem difficult. For example, certain vertices can be fixed in their partitions (fixed constraints) or the total vertex weight in each partition may be limited (balance constraints). With balance constraints, the problem of optimally partitioning a hypergraph is known to be NP-hard [28]. However, since partitioning is critical in several practical applications, heuristic algorithms were developed with near-linear runtime. Such move-based heuristics for k-way hypergraph partitioning appear in [46, 27, 14], with refinements given by [47, 58, 32, 49, 24, 10, 20, 35, 41, 25]. The following is an introduction to partitioning formulations and algorithms, centered on the Fiduccia-Mattheyses heuristic [27] and its derivatives.