A faster algorithm for finding Tarski fixed points

Introduction

Tarski’s fixed point theorem states that every order preserving function on a complete lattice has a greatest and least fixed point [Tarski55], and therefore in particular, every such function has at least one fixed point. Recently, there has been interest in the complexity of finding such a fixed point. This is due to its applications, including computing Nash equilibria of supermodular games and finding the solution of a simple stochastic game [EPRY20].

Prior work has focused on the complete lattice $L$ defined by a $k$-dimensional grid of width $n$. Dang, Qi, and Ye [DQY20] give an algorithm that finds a fixed point of a function $f:L\rightarrow L$ using $O(\log^{k}n)$ queries to $f$. This algorithm uses recursive binary search, where a $k$-dimensional problem is solved by making $\log n$ recursive calls on $(k-1)$-dimensional sub-instances. They conjectured that this algorithm is optimal.

Later work of Etessami, Papadimitriou, Rubinstein, and Yannakakis took the first step towards proving this [EPRY20]. They showed that finding a Tarski fixed point in a two-dimensional lattice requires $\Omega(\log^{2}n)$ queries, meaning that the Dang et al. algorithm is indeed optimal in the two-dimensional case. Etessami et al. conjectured that the Dang et al. algorithm is optimal for constant $k$, and they leave as an explicit open problem the question of whether their lower bound can be extended to dimension three or beyond.

Our contribution. In this paper we show that, surprisingly, the Dang et al. algorithm is not optimal in dimension three, or any higher dimension, and so we falsify the prior conjectures. We do this by giving an algorithm that can find a Tarski fixed point in three dimensions using $O(\log^{2}n)$ queries, thereby beating the $O(\log^{3}n)$ query algorithm of Dang et al.

The Dang et al. algorithm solves a three-dimensional instance by making recursive calls to find a fixed point of $\log n$ distinct two-dimensional sub-instances. Our key innovation is to point out that one does not need to find a fixed point of the two-dimensional sub-instance to make progress. Instead, we define the concept of an inner algorithm (Definition LABEL:def:inner) that, given a two-dimensional sub-instance, is permitted to return any point that lies in the up or down set of the three-dimensional instance (defined formally later). This is a much larger set of points, so whereas finding a fixed point of a two-dimensional instance requires $\Omega(\log^{2}n)$ queries [EPRY20], we give a $O(\log n)$ query inner algorithm for two-dimensional instances. This inner algorithm is quite involved, and is the main technical contribution of the paper.

We show that, given an inner algorithm for dimension $k-1$, a reasonably straightforward outer algorithm can find a Tarski fixed point by making $O(k\cdot\log n)$ calls to the inner algorithm. Thus we obtain a $O(\log^{2}n)$ query algorithm for the case where $k=3$. We leave as an open problem the question of whether efficient inner algorithms exist in higher dimensions.

For higher-dimensional instances, we show a decomposition theorem: if $a$-dimensional Tarski problems can be solved in $q_{a}$ queries, and $b$-dimensional Tarski problems can be solved in $q_{b}$ queries, then $(a\cdot b)$-dimensional Tarski can be solved in $q_{a}\cdot(q_{b}+2)$ queries. This then allows us to use our new algorithm for three-dimensional Tarski problems to obtain an algorithm that solves $k$-dimensional Tarski problems using $O(\log^{2\lceil k/3\rceil}n)$ queries, a substantial improvement over the $O(\log^{k}n)$ algorithm of Dang et al.[DQY20].

Though we state our results in terms of query complexity for the sake of simplicity, it should be pointed out that all of our algorithms run in polynomial time. Specifically, our algorithms will run in $O(\operatorname{poly}(\log n,k)\cdot\log^{2\lceil k/3\rceil}n)$ time when the function is presented as a Boolean circuit of size $\operatorname{poly}(\log n,k)$.

Related work. Etessami et al. also studied the computational complexity of the Tarski problem [EPRY20], showing that the problem lies in PPAD and PLS. However, the exact complexity of the problem remains open. It is not clear whether the problem is PPAD $\cap$ PLS-complete [FGHS20], or contained in some other lower class such as EOPL or UEOPL [FGMS20].

Tarski’s fixed point theorem has been applied in a wide range of settings within Economics [Topkis79, MilgromR90, Topkis98], and in particular to settings that can be captured by supermodular games, which are in fact equivalent to the Tarski problem [EPRY20]. In terms of algorithms, Echenique [Echenique07] studied the problem of computing all pure equilibria of a supermodular game, which is at least as hard as finding the greatest or least fixed point of the Tarski problem, which is itself NP-hard [EPRY20]. There have also been several papers that study properties of Tarski fixed points, such as the complexity of deciding whether a fixed point is unique [DQY20, DangY18, DangYe18-tech, DangY20]. The Tarski problem has also been studied in the setting where the partial order is given by an oracle [ChangLT08].

Preliminaries

The Tarski fixed point problem. Given a lattice $L$, a function $f:L\rightarrow L$ is order preserving if $f(x)\preceq f(y)$ whenever $x\preceq y$. A point $x\in L$ is fixed point of $f$ if $f(x)=x$. A weak version of Tarski’s theorem can be stated as follows.

Every order preserving function on a complete lattice has a fixed point.

Thus, we can define a total search problem for Tarski’s fixed point theorem.

Given a lattice $L$, and a function $f:L\rightarrow L$, find one of:

Two points $x,y\in L$ such that $x\preceq y$ and $f(x)\not\preceq f(y)$.

Solutions of type (T1) are fixed points of $f$, whereas solutions of type (T2) witness that $f$ is not an order preserving function. By Tarski’s theorem, if a function $f$ has no solutions of type (T2), then it must have a solution of type (T1), and so

The left-hand picture in Figure 2.6 gives an example of a two-dimensional