Improved Upper Bounds for Finding Tarski Fixed Points
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We study the query complexity of finding a Tarski fixed point over the k-dimensional grid {1,…,n}^k. Improving on the previous best upper bound of O(log^⌈ 2k/3⌉ n) [FPS20], we give a new algorithm with query complexity O(log^⌈ (k+1)/2⌉ n). This is based on a novel decomposition theorem about a weaker variant of the Tarski fixed point problem, where the input consists of a monotone function f:[n]^k→ [n]^k and a monotone sign function b:[n]^k→{-1,0,1} and the goal is to find an x∈ [n]^k that satisfies either f(x)≼ x and b(x)≤ 0 or f(x)≽ x and b(x)≥ 0.
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