On constrained matchings, stable under random preferences
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929387143692288 |
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| author | Pittel, Boris |
| author_facet | Pittel, Boris |
| contents | Colloquially, there are two groups, $n$ men and $n$ women, each man (woman) ranking women (men) as potential marriage partners. A complete matching is called stable if no unmatched pair prefer each other to their partners in the matching. If some pairs are not admissible, then such a matching may not exist, but a properly defined partial stable matching exists always, and all such matchings involve the same, equi-numerous, groups of men and women. Earlier we proved that, for the complete, random, preference lists, with high probability (whp) the total number of complete stable matchings is, roughly, of order $n^{1/2}$, at least. Here we consider the case that the preference lists are still complete, but a generic pair (man,woman) is admissible with probability $p$, independently of all other $n^2-1$ pairs. It is shown that the expected number of complete stable matchings tends to $0$ if, roughly, $p<\tfrac{\log^2 n}{n}$ and to infinity if $p>\tfrac{\log^2 n}{n}$. We show that whp: (a) there exists a complete stable matching if $p>(9/4)\tfrac{\log^2 n}{n}$, (b) the number of unmatched men and women is bounded if $p> \tfrac{\log^2n}{n}$, and (c) this number grows as a fractional power of $n$ for $p<\tfrac{\log^2 n}{n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_10319 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On constrained matchings, stable under random preferences Pittel, Boris Combinatorics 60C05, 05C05, 92B10 Colloquially, there are two groups, $n$ men and $n$ women, each man (woman) ranking women (men) as potential marriage partners. A complete matching is called stable if no unmatched pair prefer each other to their partners in the matching. If some pairs are not admissible, then such a matching may not exist, but a properly defined partial stable matching exists always, and all such matchings involve the same, equi-numerous, groups of men and women. Earlier we proved that, for the complete, random, preference lists, with high probability (whp) the total number of complete stable matchings is, roughly, of order $n^{1/2}$, at least. Here we consider the case that the preference lists are still complete, but a generic pair (man,woman) is admissible with probability $p$, independently of all other $n^2-1$ pairs. It is shown that the expected number of complete stable matchings tends to $0$ if, roughly, $p<\tfrac{\log^2 n}{n}$ and to infinity if $p>\tfrac{\log^2 n}{n}$. We show that whp: (a) there exists a complete stable matching if $p>(9/4)\tfrac{\log^2 n}{n}$, (b) the number of unmatched men and women is bounded if $p> \tfrac{\log^2n}{n}$, and (c) this number grows as a fractional power of $n$ for $p<\tfrac{\log^2 n}{n}$. |
| title | On constrained matchings, stable under random preferences |
| topic | Combinatorics 60C05, 05C05, 92B10 |
| url | https://arxiv.org/abs/2406.10319 |