Testability of relations between permutations

TL;DR

We address the problem of testing relations between permutations by developing a graph-encoded framework in which a system $E$ corresponds to a graph family $\mathrm{GSol}_E$ and testability is governed by expansion properties of these graphs. The main tool is translating algebraic constraints into isoperimetric/expansion criteria, connecting to group-theoretic notions via a derived group $\Gamma(E)$ and enabling broad applicability. The key results show that if every graph in $\mathrm{GSol}_E$ is nonexpanding ($\iota(G)=0$), then $E$ is testable, while a positive Cheeger constant lower bound across the finite connected graphs $\mathrm{FGSol}_E$ that remain infinite implies non-testability. The work also reframes stability in permutations as a special case of testability, surveys prior computational results, and outlines future directions for complete characterizations and broader applications.

Abstract

We initiate the study of property testing problems concerning relations between permutations. In such problems, the input is a tuple $(σ_1,\dotsc,σ_d)$ of permutations on $\{1,\dotsc,n\}$, and one wishes to determine whether this tuple satisfies a certain system of relations $E$, or is far from every tuple that satisfies $E$. 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 relation $\mathsf{XY=YX}$, this corresponds to testing whether $σ_1σ_2=σ_2σ_1$, where $σ_1σ_2$ and $σ_2σ_1$ denote composition of permutations. We define a collection of graphs, naturally associated with the system $E$, that encodes all the information relevant to the testability of $E$. We then prove two theorems that provide criteria for testability and non-testability in terms of expansion properties of these graphs. By virtue of a deep connection with group theory, both theorems are applicable to wide classes of systems of relations. In addition, we formulate the well-studied group-theoretic notion of stability in permutations as a special case of the testability notion above, interpret all previous works on stability as testability results, survey previous results on stability from a computational perspective, and describe many directions for future research on stability and testability.

Theorems & Definitions (10)

• Definition 1.1: Measures of expansion
• Theorem 1: Main positive theorem
• Theorem 2: Main negative theorem
• Definition 1.2
• Definition 1.3
• Example 1.4
• Definition 1.5
• Definition 1.6: Testable system of relations
• Definition 1.7
• Example 1.8