Statistical estimation for optimization problems on graphs
This is an incremental position paper that aims to encourage further research into statistical estimation for graph-based optimization problems, relevant for machine learning and data mining applications.
The paper tackles the problem of estimating summary statistics for combinatorial optimization problems on graphs, such as expected shortest path lengths or minimum spanning tree weights, by proposing a statistical estimation framework focused on spanning trees as a concrete example.
Large graphs abound in machine learning, data mining, and several related areas. A useful step towards analyzing such graphs is that of obtaining certain summary statistics - e.g., or the expected length of a shortest path between two nodes, or the expected weight of a minimum spanning tree of the graph, etc. These statistics provide insight into the structure of a graph, and they can help predict global properties of a graph. Motivated thus, we propose to study statistical properties of structured subgraphs (of a given graph), in particular, to estimate the expected objective function value of a combinatorial optimization problem over these subgraphs. The general task is very difficult, if not unsolvable; so for concreteness we describe a more specific statistical estimation problem based on spanning trees. We hope that our position paper encourages others to also study other types of graphical structures for which one can prove nontrivial statistical estimates.