An Optimal Randomized Algorithm for Finding the Saddlepoint
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arXiv
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| Format: | Preprint |
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2024
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| author | Dallant, Justin Haagensen, Frederik Jacob, Riko Kozma, László Wild, Sebastian |
| author_facet | Dallant, Justin Haagensen, Frederik Jacob, Riko Kozma, László Wild, Sebastian |
| contents | A \emph{saddlepoint} of an $n \times n$ matrix is an entry that is the maximum of its row and the minimum of its column. Saddlepoints give the \emph{value} of a two-player zero-sum game, corresponding to its pure-strategy Nash equilibria; efficiently finding a saddlepoint is thus a natural and fundamental algorithmic task.
For finding a \emph{strict saddlepoint} (an entry that is the strict maximum of its row and the strict minimum of its column) we recently gave an $O({n\log^*{n}})$-time algorithm, improving the $O({n\log{n}})$ bounds from 1991 of Bienstock, Chung, Fredman, Schäffer, Shor, Suri and of Byrne and Vaserstein.
In this paper we present an optimal $O({n})$-time algorithm for finding a strict saddlepoint based on random sampling. Our algorithm, like earlier approaches, accesses matrix entries only via unit-cost binary comparisons. For finding a (non-strict) saddlepoint, we extend an existing lower bound to randomized algorithms, showing that the trivial $O(n^2)$ runtime cannot be improved even with the use of randomness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Optimal Randomized Algorithm for Finding the Saddlepoint Dallant, Justin Haagensen, Frederik Jacob, Riko Kozma, László Wild, Sebastian Computational Complexity Data Structures and Algorithms Combinatorics F.2.0 A \emph{saddlepoint} of an $n \times n$ matrix is an entry that is the maximum of its row and the minimum of its column. Saddlepoints give the \emph{value} of a two-player zero-sum game, corresponding to its pure-strategy Nash equilibria; efficiently finding a saddlepoint is thus a natural and fundamental algorithmic task. For finding a \emph{strict saddlepoint} (an entry that is the strict maximum of its row and the strict minimum of its column) we recently gave an $O({n\log^*{n}})$-time algorithm, improving the $O({n\log{n}})$ bounds from 1991 of Bienstock, Chung, Fredman, Schäffer, Shor, Suri and of Byrne and Vaserstein. In this paper we present an optimal $O({n})$-time algorithm for finding a strict saddlepoint based on random sampling. Our algorithm, like earlier approaches, accesses matrix entries only via unit-cost binary comparisons. For finding a (non-strict) saddlepoint, we extend an existing lower bound to randomized algorithms, showing that the trivial $O(n^2)$ runtime cannot be improved even with the use of randomness. |
| title | An Optimal Randomized Algorithm for Finding the Saddlepoint |
| topic | Computational Complexity Data Structures and Algorithms Combinatorics F.2.0 |
| url | https://arxiv.org/abs/2401.06512 |