Graph Pooling with Maximum-Weight $k$-Independent Sets

Graph reductions are fundamental when dealing with large scale networks and relational data. They allow to downsize tasks of high computational impact by solving them in coarsened structures. At the same time, graph reductions play the role of pooling layers in graph neural networks, to extract multi-resolution representations from structures. In these contexts, the ability of the reduction mechanism to preserve distance relationships and topological properties appears fundamental, along with a scalability enabling its application to real-world sized problems. In this paper, we introduce a graph coarsening mechanism based on the graph-theoretic concept of maximum-weight $k$-independent sets, providing a greedy algorithm that allows efficient parallel implementation on GPUs. Our method is the first graph-structured counterpart of controllable equispaced coarsening mechanisms in regular data (images, sequences). We prove theoretical guarantees for distortion bounds on path lengths, as well as the ability to preserve key topological properties in the coarsened graphs. We leverage these concepts to define a graph pooling mechanism that we empirically assess in graph classification tasks, showing that it compares favorably against pooling methods in literature.