Stability and testability: equations in permutations

11/10/2020
by   Oren Becker, et al.
0

We initiate the study of property testing problems concerning equations in permutations. In such problems, the input consists of permutations σ_1,…,σ_d∈Sym(n), and one wishes to determine whether they satisfy a certain system of equations E, or are far from doing so. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that E is testable. For example, when d=2 and E consists of the single equation 𝖷𝖸=𝖸𝖷, this corresponds to testing whether σ_1σ_2=σ_2σ_1. We formulate the well-studied group-theoretic notion of stability in permutations as a testability concept, and interpret all works on stability as testability results. Furthermore, we establish a close connection between testability and group theory, and harness the power of group-theoretic notions such as amenability and property (T) to produce a large family of testable equations, beyond those afforded by the study of stability, and a large family of non-testable equations. Finally, we provide a survey of results on stability from a computational perspective and describe many directions for future research.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

04/09/2022

Testability in group theory

This paper is a journal counterpart to our FOCS 2021 paper, in which we ...
12/21/2018

Testing Stability Properties in Graphical Hedonic Games

In hedonic games, players form coalitions based on individual preference...
01/29/2021

On the relation of powerflow and Telegrapher's equations: continuous and numerical Lyapunov stability

In this contribution we analyze the exponential stability of power netwo...
05/11/2020

Stability-enhanced AP IMEX1-LDG method: energy-based stability and rigorous AP property

In our recent work [22], a family of high order asymptotic preserving (A...
03/01/2021

Structure-preserving numerical schemes for Lindblad equations

We study a family of structure-preserving deterministic numerical scheme...
03/24/2013

On Learnability, Complexity and Stability

We consider the fundamental question of learnability of a hypotheses cla...
05/16/2019

Stability of Linear Structural Equation Models of Causal Inference

We consider the numerical stability of the parameter recovery problem in...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.