A concentration inequality for random combinatorial optimisation problems
Preprint, 2024
Given a finite set S, i.i.d. random weights Xᵢ, i∈S, and a family of subsets F⊆2^S, we consider the minimum weight of an F∈F:
M(F):= min { Σ_{i∈F} Xᵢ: F∈F}
In particular, we investigate under what conditions this random variable is sharply concentrated around its mean.
We define the patchability of a family F: essentially, how expensive is it to finish an almost-complete F (that is, F is close to F in Hamming distance) if the edge weights are re-randomized?
Combining the patchability of F, applying the Talagrand inequality to a dual problem, and a sprinkling-type argument, we prove a concentration inequality for the random variable M(F).
M(F):= min { Σ_{i∈F} Xᵢ: F∈F}
In particular, we investigate under what conditions this random variable is sharply concentrated around its mean.
We define the patchability of a family F: essentially, how expensive is it to finish an almost-complete F (that is, F is close to F in Hamming distance) if the edge weights are re-randomized?
Combining the patchability of F, applying the Talagrand inequality to a dual problem, and a sprinkling-type argument, we prove a concentration inequality for the random variable M(F).
Author
Joel Danielsson
Chalmers, Mathematical Sciences, Analysis and Probability Theory
Subject Categories (SSIF 2011)
Probability Theory and Statistics
DOI
10.48550/ARXIV.2407.12672