A concentration inequality for random combinatorial optimisation problems

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 $F\in \mathcal{F}$: $$M(\mathcal{F}):=\underset{F\in \mathcal{F}}{\min }\sum _{i\in F}{X}_i\mathrm{.}$$ 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 $M(\mathcal{F})$.