Improved Upper Bounds for Finding Tarski Fixed Points

TL;DR

This paper presents an improved algorithm for finding Tarski fixed points in k-dimensional grids, reducing query complexity by leveraging a new decomposition theorem for a related monotone function problem.

Contribution

It introduces a novel decomposition theorem and an improved algorithm that lowers the query complexity for finding Tarski fixed points over grids.

Findings

Query complexity improved to O(log^{ceil((k+1)/2)} n)

New decomposition theorem for a weaker Tarski fixed point variant

Enhanced understanding of monotone function problems

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 $b(x)\ge 0$.