ﻻ يوجد ملخص باللغة العربية
We initiate the study of property testing problems concerning relations between permutations. In such problems, the input is a tuple $(sigma_1,dotsc,sigma_d)$ 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 $sigma_1sigma_2=sigma_2sigma_1$, where $sigma_1sigma_2$ and $sigma_2sigma_1$ 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.
Given an abelian group $G$, it is natural to ask whether there exists a permutation $pi$ of $G$ that destroys all nontrivial 3-term arithmetic progressions (APs), in the sense that $pi(b) - pi(a) eq pi(c) - pi(b)$ for every ordered triple $(a,b,c) i
In the 1970s, Lovasz built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings (FCT 1979). A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the non-co
In this work we show that high dimensional expansion implies locally testable code. Specifically, we define a notion that we call high-dimensional-expanding-system (HDE-system). This is a set system defined by incidence relations with certain high di
We consider finite sums of counting functions on the free group $F_n$ and the free monoid $M_n$ for $n geq 2$. Two such sums are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivale
We study the multiparty communication complexity of high dimensional permutations, in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where thr