Tutte Colloquium - Chandra Chekuri

Title: Densest Subgraph: Supermodularity, Iterative Peeling, and Flow

Abstract:

The densest subgraph problem in a graph (DSG), in the simplest form, is the following. Given an undirected graph G = (V, E) find a subset S of vertices that maximizes the ratio |E(S)|/|S| where E(S) is the set of edges with both endpoints in S. DSG and several of its variants are well-studied in theory and practice and have many applications in data mining and network analysis. We study fast algorithms and
structural aspects of DSG via the lens of supermodularity. For this we consider the densest supermodular
subset problem (DSS): given a non-negative supermodular function f over a ground set V, maximize f(S)/|S|.

For DSG we describe a simple flow-based algorithm that outputs a (1−\epsilon)-approximation in deterministic O(m polylog(n)/\epsilon) time where m is the number of edges.

Greedy peeling algorithms have been very popular for DSG and several variants due to their efficiency,
empirical performance, and worst-case approximation guarantees. We describe a simple peeling algorithm for DSS and analyze its approximation guarantee in a fashion that unifies several existing results. Boob et al.
developed an iterative peeling algorithm for DSG which appears to work very well in practice, and made a
conjecture about its convergence to optimality. We affirmatively answer their conjecture, and in fact prove
that a natural generalization of their algorithm converges for any supermodular function f; the key to our proof is to consider an LP formulation that is derived via the Lovász extension of a supermodular function which is inspired by the LP relaxation of Charikar that has played an important role in several developments.

This is joint work with Kent Quanrud and Manuel Torres. The paper appeared in ACM-SIAM SODA 2022 and is available at http://chekuri.cs.illinois.edu/papers/densest-subgraph-soda22.pdf