A concentration inequality for random combinatorial optimisation problems
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913434503741440 |
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| author | Danielsson, Joel Larsson |
| author_facet | Danielsson, Joel Larsson |
| contents | 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 $M(\mathcal{F})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12672 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A concentration inequality for random combinatorial optimisation problems Danielsson, Joel Larsson Combinatorics Probability 60C05 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 $M(\mathcal{F})$. |
| title | A concentration inequality for random combinatorial optimisation problems |
| topic | Combinatorics Probability 60C05 |
| url | https://arxiv.org/abs/2407.12672 |