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Improved Upper Bounds for Finding Tarski Fixed Points (2202.05913v2)
Published 11 Feb 2022 in cs.GT and cs.CC
Abstract: We study the query complexity of finding a Tarski fixed point over the $k$-dimensional grid ${1,\ldots,n}k$. Improving on the previous best upper bound of $\smash{O(\log{\lceil 2k/3\rceil} n)}$ [FPS20], we give a new algorithm with query complexity $\smash{O(\log{\lceil (k+1)/2\rceil} n)}$. 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 $f:[n]k\rightarrow [n]k$ and a monotone sign function $b:[n]k\rightarrow {-1,0,1}$ and the goal is to find an $x\in [n]k$ that satisfies $either$ $f(x)\preceq x$ and $b(x)\le 0$ $or$ $f(x)\succeq x$ and $b(x)\ge 0$.