Secretary, Prophet, and Stochastic Probing via Big-Decisions-First
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911559375126528 |
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| author | Rubinstein, Aviad Singla, Sahil |
| author_facet | Rubinstein, Aviad Singla, Sahil |
| contents | We revisit three fundamental problems in algorithms under uncertainty: the Secretary Problem, Prophet Inequality, and Stochastic Probing, each subject to general downward-closed constraints. When elements have binary values, all three problems admit a tight $\tildeΘ(\log n)$-factor approximation guarantee. For general (non-binary) values, however, the best known algorithms lose an additional $\log n$ factor when discretizing to binary values, leaving a quadratic gap of $\tildeΘ(\log n)$ vs. $\tildeΘ(\log^2 n)$.
We resolve this quadratic gap for all three problems, showing $\tildeΩ(\log^2 n)$-hardness for two of them and an $O(\log n)$-approximation algorithm for the third. While the technical details differ across settings, and between algorithmic and hardness proofs, all our results stem from a single core observation, which we call the Big-Decisions-First Principle: Under uncertainty, it is better to resolve high-stakes (large-value) decisions early. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00437 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Secretary, Prophet, and Stochastic Probing via Big-Decisions-First Rubinstein, Aviad Singla, Sahil Data Structures and Algorithms Computer Science and Game Theory We revisit three fundamental problems in algorithms under uncertainty: the Secretary Problem, Prophet Inequality, and Stochastic Probing, each subject to general downward-closed constraints. When elements have binary values, all three problems admit a tight $\tildeΘ(\log n)$-factor approximation guarantee. For general (non-binary) values, however, the best known algorithms lose an additional $\log n$ factor when discretizing to binary values, leaving a quadratic gap of $\tildeΘ(\log n)$ vs. $\tildeΘ(\log^2 n)$. We resolve this quadratic gap for all three problems, showing $\tildeΩ(\log^2 n)$-hardness for two of them and an $O(\log n)$-approximation algorithm for the third. While the technical details differ across settings, and between algorithmic and hardness proofs, all our results stem from a single core observation, which we call the Big-Decisions-First Principle: Under uncertainty, it is better to resolve high-stakes (large-value) decisions early. |
| title | Secretary, Prophet, and Stochastic Probing via Big-Decisions-First |
| topic | Data Structures and Algorithms Computer Science and Game Theory |
| url | https://arxiv.org/abs/2604.00437 |