Coarsening at random is a technique used in the field of computer science, particularly in the domain of graph algorithms. It involves simplifying a complex graph by merging together or “coarsening” pairs of vertices in the graph. This can be useful in a number of scenarios, such as reducing the size of the graph to make it more manageable, or improving the efficiency of certain algorithms that are applied to the graph.

One example of coarsening at random is the “contraction algorithm” for finding minimum cuts in a graph. Given a graph with vertices and edges, the contraction algorithm proceeds as follows:

Pick a pair of vertices in the graph at random and merge them together, resulting in a new, coarsened graph with one fewer vertex.

Repeat this process until only two vertices remain in the graph.

The minimum cut of the original graph is then defined as the set of edges that were removed during the coarsening process.

This algorithm can be used to solve a wide range of problems, such as finding the minimum-cost flow in a network, or determining the optimal configuration of a circuit.

Another example of coarsening at random is the “multilevel algorithm” for graph partitioning. Given a graph with vertices and edges, the multilevel algorithm proceeds as follows:

Coarsen the graph by merging pairs of vertices at random until the graph has a small number of vertices (e.g. 10-20% of the original number of vertices).

Use a graph partitioning algorithm (such as Kernighan-Lin or Fiduccia-Mattheyses) to partition the coarsened graph into a number of clusters.

Uncoarsen the graph by repeatedly splitting each cluster in the coarsened graph into two sub-clusters, until the original graph is obtained.

Use the partitioning of the coarsened graph as a starting point for refining the partitioning of the original graph.

The multilevel algorithm can be used to solve a wide range of problems, such as partitioning a circuit for efficient layout, or dividing a computer network into clusters for improved communication.

In general, coarsening at random can be an effective technique for simplifying complex graphs and improving the efficiency of graph algorithms. It allows for a fast, approximate solution to be obtained, which can then be refined using more computationally intensive methods.