Edge-Colored Clustering in Hypergraphs: Beyond Minimizing Unsatisfied Edges

We consider a framework for clustering edge-colored hypergraphs, where the goal is to cluster (equivalently, to color) objects based on the primary type of multiway interactions they participate in. One well-studied objective is to color nodes to minimize the number of unsatisfied hyperedges---those containing one or more nodes whose color does not match the hyperedge color. We motivate and present advances for several directions that extend beyond this minimization problem. We first provide new algorithms for maximizing satisfied edges, which is the same at optimality but is much more challenging to approximate, with all prior work restricted to graphs. We develop the first approximation algorithm for hypergraphs, and then refine it to improve the best-known approximation factor for graphs. We then introduce new objective functions that incorporate notions of balance and fairness, and provide new hardness results, approximations, and fixed-parameter tractability results.

Clustering is the fundamental computational task of partitioning a dataset into groups of similar data objects. We present new algorithms and complexity results for a framework for clustering objects based on categorical, multiway interactions. For example, in a dataset of workers, prior team assignments can be modeled as multiway interactions with categorical labels (where the category can indicate the type of task that team worked on). Here, our clustering framework could be used to assign workers to future tasks based on prior experience. Among other results, we present new approaches for ensuring the output clustering satisfies certain balance requirements with respect to different categories of interactions. In our example, one such balance condition would be that workers are assigned to task types in such a way that certain types of tasks are not favored far more than others.