A concentration inequality for random combinatorial optimisation problems

TL;DR

This paper establishes a concentration inequality for the minimum weight of subsets in a family under i.i.d. random weights, using patchability, Talagrand inequality, and a sprinkling argument to analyze sharp concentration.

Contribution

It introduces the concept of patchability and applies advanced probabilistic tools to derive concentration bounds for combinatorial optimization problems.

Findings

Proves a concentration inequality for minimum subset weights.

Defines the patchability property of set families.

Demonstrates sharp concentration under certain conditions.

Abstract

Given a finite set $S$, i.i.d. random weights $\{X_i\}_{i\in S}$, and a family of subsets $\mathcal{F}\subseteq 2^S$, we consider the minimum weight of an $F\in \mathcal{F}$: \[ M(\mathcal{F}):= \min_{F\in \mathcal{F}} \sum_{i\in F}X_i. \] In particular, we investigate under what conditions this random variable is sharply concentrated around its mean. We define the patchability of a family $\mathcal{F}$: essentially, how expensive is it to finish an almost-complete $F$ (that is, $F$ is close to $\mathcal{F}$ in Hamming distance) if the edge weights are re-randomized? Combining the patchability of $\mathcal{F}$, applying the Talagrand inequality to a dual problem, and a sprinkling-type argument, we prove a concentration inequality for the random variable $M(\mathcal{F})$.