Matroid Secretary via Labeling Schemes
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912404576665600 |
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| author | Bérczi, Kristóf Livanos, Vasilis Soto, José Verdugo, Victor |
| author_facet | Bérczi, Kristóf Livanos, Vasilis Soto, José Verdugo, Victor |
| contents | The Matroid Secretary Problem (MSP) is one of the most prominent settings for online resource allocation and optimal stopping. A decision-maker is presented with a ground set of elements $E$ revealed sequentially and in random order. Upon arrival, an irrevocable decision is made in a take-it-or-leave-it fashion, subject to a feasibility constraint on the set of selected elements captured by a matroid defined over $E$. The decision-maker only has ordinal access to compare the elements, and the goal is to design an algorithm that selects every element of the optimal basis with probability at least $α$ (i.e., $α$-probability-competitive). While the existence of a constant probability-competitive algorithm for MSP remains a major open question, simple greedy policies are at the core of state-of-the-art algorithms for several matroid classes.
We introduce a flexible and general algorithmic framework to analyze greedy-like algorithms for MSP based on constructing a language associated with the matroid. Using this language, we establish a lower bound on the probability-competitiveness of the algorithm by studying a corresponding Poisson point process that governs the words' distribution in the language. Using our framework, we break the state-of-the-art guarantee for laminar matroids by settling the probability-competitiveness of the greedy-improving algorithm to be exactly $1-\ln(2) \approx 0.3068$. We also showcase the capabilities of our framework in graphic matroids, to show a probability-competitiveness of $0.2693$ for simple graphs and $0.2504$ for general graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12069 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matroid Secretary via Labeling Schemes Bérczi, Kristóf Livanos, Vasilis Soto, José Verdugo, Victor Data Structures and Algorithms The Matroid Secretary Problem (MSP) is one of the most prominent settings for online resource allocation and optimal stopping. A decision-maker is presented with a ground set of elements $E$ revealed sequentially and in random order. Upon arrival, an irrevocable decision is made in a take-it-or-leave-it fashion, subject to a feasibility constraint on the set of selected elements captured by a matroid defined over $E$. The decision-maker only has ordinal access to compare the elements, and the goal is to design an algorithm that selects every element of the optimal basis with probability at least $α$ (i.e., $α$-probability-competitive). While the existence of a constant probability-competitive algorithm for MSP remains a major open question, simple greedy policies are at the core of state-of-the-art algorithms for several matroid classes. We introduce a flexible and general algorithmic framework to analyze greedy-like algorithms for MSP based on constructing a language associated with the matroid. Using this language, we establish a lower bound on the probability-competitiveness of the algorithm by studying a corresponding Poisson point process that governs the words' distribution in the language. Using our framework, we break the state-of-the-art guarantee for laminar matroids by settling the probability-competitiveness of the greedy-improving algorithm to be exactly $1-\ln(2) \approx 0.3068$. We also showcase the capabilities of our framework in graphic matroids, to show a probability-competitiveness of $0.2693$ for simple graphs and $0.2504$ for general graphs. |
| title | Matroid Secretary via Labeling Schemes |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2411.12069 |