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View all- Fearnley JGoldberg PHollender ASavani R(2022)The Complexity of Gradient Descent: CLS = PPAD ∩ PLSJournal of the ACM10.1145/356816370:1(1-74)Online publication date: 19-Dec-2022
A Faster Algorithm for Finding Tarski Fixed Points
Article No.: 23, Pages 1 - 23
Abstract
Dang et al. have given an algorithm that can find a Tarski fixed point in a k-dimensional lattice of width n using O(log k n) queries [2]. Multiple authors have conjectured that this algorithm is optimal [2, 7], and indeed this has been proven for two-dimensional instances [7]. We show that these conjectures are false in dimension three or higher by giving an O(log2 n) query algorithm for the three-dimensional Tarski problem. We also give a new decomposition theorem for k-dimensional Tarski problems which, in combination with our new algorithm for three dimensions, gives an O(log2 ⌈k/3⌉ n) query algorithm for the k-dimensional problem.
References
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Chuangyin Dang, Qi Qi, and Yinyu Ye. 2020. Computations and complexities of Tarski’s fixed points and supermodular games. CoRR abs/2005.09836 (2020). Stanford Tech Report version appeared in 2012.
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Chuangyin Dang and Yinyu Ye. 2018. On the Complexity of a Class of Discrete Fixed Point Problems Under the Lexicographic Ordering. Technical Report. City University of Hong Kong, CY2018-3, 17 pages.
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Chuangyin Dang and Yinyu Ye. 2018. On the complexity of an expanded Tarski’s fixed point problem under the componentwise ordering. Theor. Comput. Sci. 732 (2018), 26–45.
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Chuangyin Dang and Yinyu Ye. 2020. Erratum/correction to “On the complexity of an expanded Tarski’s fixed point problem under the componentwise ordering” [Theor. Comput. Sci. 732 (2018) 26–45]. Theor. Comput. Sci. 817 (2020), 80.
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Index Terms
- A Faster Algorithm for Finding Tarski Fixed Points
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Publication History
Published: 11 October 2022
Online AM: 17 March 2022
Accepted: 02 February 2022
Received: 15 June 2021
Published in TALG Volume 18, Issue 3
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View all- Fearnley JGoldberg PHollender ASavani R(2022)The Complexity of Gradient Descent: CLS = PPAD ∩ PLSJournal of the ACM10.1145/356816370:1(1-74)Online publication date: 19-Dec-2022
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