Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Mystery Deepens: On the Query Complexity of Tarski Fixed Points

Xi Chen, Yuhao Li, Mihalis Yannakakis

Abstract

We give an $O(\log^2 n)$-query algorithm for finding a Tarski fixed point over the $4$-dimensional lattice $[n]^4$, matching the $Ω(\log^2 n)$ lower bound of [EPRY20]. Additionally, our algorithm yields an ${O(\log^{\lceil (k-1)/3\rceil+1} n)}$-query algorithm for any constant $k$, improving the previous best upper bound ${O(\log^{\lceil (k-1)/2\rceil+1} n)}$ of [CL22]. Our algorithm uses a new framework based on \emph{safe partial-information} functions. The latter were introduced in [CLY23] to give a reduction from the Tarski problem to its promised version with a unique fixed point. This is the first time they are directly used to design new algorithms for Tarski fixed points.

The Mystery Deepens: On the Query Complexity of Tarski Fixed Points

Abstract

We give an -query algorithm for finding a Tarski fixed point over the -dimensional lattice , matching the lower bound of [EPRY20]. Additionally, our algorithm yields an -query algorithm for any constant , improving the previous best upper bound of [CL22]. Our algorithm uses a new framework based on \emph{safe partial-information} functions. The latter were introduced in [CLY23] to give a reduction from the Tarski problem to its promised version with a unique fixed point. This is the first time they are directly used to design new algorithms for Tarski fixed points.

Paper Structure

This paper contains 25 sections, 28 theorems, 58 equations, 6 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Prior Work
  3. Our Contributions
  4. Additional Related Work
  5. Technical Overview
  6. Candidate Sets of Monotone PI functions
  7. Safe PI Functions
  8. The Safe PI Function Game
  9. Our Construction of Candidate Sets
  10. Trimming Candidate Sets When $k=3$
  11. Preliminaries
  12. Simple Functions and Sign Functions
  13. Partial-Information Functions
  14. Safe PI Functions and Safe Functions
  15. The Safe PI Function Game
  16. ...and 10 more sections

Key Result

Theorem 1

There is an $O(\log^2n)$-query algorithm for $\textsc{Tarski}(n,4)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Illustrations of monotone PI functions. Suppose that ${g:[8]^2\mapsto\left \{ -1,0,1 \right \} ^2\times\left \{ \pm 1 \right \} }$ is a monotone function. On $x\in[8]^2$, a blue right arrow means $g(x)_1=+1$, a blue down arrow means $g(x)_2=-1$, a green "$+$" means $g(x)_3=+1$, and (for figures below) a green "$-$" means $g(x)_3=-1$. In this and subsequent figures, we omit entries of $\{\ge,\leq\}$ in the partial information for clarity.
  • Figure 2: Illustrations of candidate sets. The red region in each figure is a candidate set of the given monotone PI function, respectively.
  • Figure 3: Illustrations for the three examples of candidate sets.
  • Figure 4: An example in which the intersection of $\texttt{Cand}_{[k]}(p)$ and $\texttt{Cand}_{k+1}(p)$ fails.
  • Figure 5: Illustrations of $\texttt{Cand}_{[k]}(p)$, $\texttt{Cand}_{k+1}(p)$ and the final $\texttt{Cand}(p)$.
  • ...and 1 more figures

Theorems & Definitions (66)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Definition 1: Monotone Functions for $\textsc{Tarski}^*(n,k)$
  • Definition 2: Slices
  • Definition 3: Consistency
  • Definition 4: Monotone PI Functions
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 56 more