## Graph and Hypergraph Sparsification

Luca Trevisan

Bocconi University

April 27, 2020

A weighted graph H is a sparsifier of a graph G if H has much fewer edges than G and, in an appropriate technical sense, H "approximates" G. Sparsifiers are useful as compressed representations of graphs and to speed up certain graph algorithms. In a "cut sparsifier," the notion of approximation is that every cut is crossed by approximately the same number of edges in G as in H. In a "spectral sparsifier" a stronger, linear-algebraic, notion of approximation holds. Similar definitions can be given for hypergraphs.