Large deviations of the greedy independent set algorithm on sparse random graphs

Combinatorics Seminar
Friday, January 15, 2021 - 3:00pm for 1 hour (actually 50 minutes)
Location (To receive the password, please email Lutz Warnke)
Brett Kolesnik – University of California, Berkeley –
Lutz Warnke

We study the greedy independent set algorithm on sparse Erdős-Rényi random graphs G(n,c/n). This range of p is of interest due to the threshold at c=e, beyond which it appears that greedy algorithms are affected by a sudden change in the independent set landscape. A large deviation principle was recently established by Bermolen et al. (2020), however, the proof and rate function are somewhat involved. Upper bounds for the rate function were obtained earlier by Pittel (1982). By discrete calculus, we identify the optimal trajectory realizing a given large deviation and obtain the rate function in a simple closed form. In particular, we show that Pittel's bounds are sharp. The proof is brief and elementary. We think the methods presented here will be useful in analyzing the tail behavior of other random growth and exploration processes.

Based on