Improved Upper Bounds for Finding Tarski Fixed Points

Abstract

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)[7], 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 a point x ∈ [n]k that satisfies either f(x) ≼ x and b(x) ≤ 0 or f(x) ≽ x and b(x) ≥ 0.