Large deviations of the greedy independent set algorithm on sparse random graphs. Issue 2 (14th December 2021)

• Record Type: Journal Article
• Authors: Kolesnik, Brett
• Is Part Of: Random structures & algorithms. Volume 61:Issue 2(2022)
• Page Start: 353
• Page End: 363
• Publication Date: 2021-12-14
• Subjects: discrete calculus -- Euler–Lagrange equation -- greedy algorithm -- independent set -- large deviations -- random graph; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.21064
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: We study the greedy independent set algorithm on sparse Erdős–Rényi random graphs G ( n, c / n ) . A large deviation principle was recently established by Bermolen et al. in 2020, however, the proof and rate function are somewhat involved. Upper bounds for the rate function were obtained earlier by Pittel in 1982. Using 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 processes.