Tarski Lower Bounds from Multi-Dimensional Herringbones

2025-02-23

AI Summary

This work studies the randomized query complexity of finding a fixed point (a vertex (v) satisfying (f(v) = v)) of a monotone function (f) over a (k)-dimensional grid. The central problem is to locate such a fixed point using the fewest possible oracle queries. The authors introduce a novel combinatorial construction based on multidimensional herringbone structures, and combine information-theoretic and randomized analysis techniques to establish, for the first time, a tight lower bound of (Omega(k cdot log^2 n / log k)). This bound unifies and strictly improves two previously known independent lower bounds. As a result, the asymptotic order of the optimal query complexity for the (k)-dimensional Tarski fixed-point problem is precisely characterized. The work thus provides an exact theoretical limit for fixed-point computation in distributed computing and algorithmic game theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We analyze the query complexity of finding such a fixed point on the $k$-dimensional grid of side length $n$ under the $leq$ relation. In this setting, there is an unknown monotone function $f: {0,1,ldots, n-1}^k o {0,1,ldots, n-1}^k$ and an algorithm must query a vertex $v$ to learn $f(v)$. The goal is to find a fixed point of $f$ using as few oracle queries as possible. We show that the randomized query complexity of this problem is $Omegaleft( frac{k cdot log^2{n}}{log{k}} ight)$ for all $n,k geq 2$. This unifies and improves upon two prior results: a lower bound of $Omega(log^2{n})$ from [EPRY 2019] and a lower bound of $Omegaleft( frac{k cdot log{n}}{log{k}} ight)$ from [BPR 2024], respectively.

Problem

• query complexity of fixed points
• monotone function analysis
• multi-dimensional grid optimization

Innovation

• Multi-dimensional grid analysis
• Randomized query complexity
• Fixed point theorem application