Hereditary properties of permutations are strongly testable

We show that for every hereditary permutation property and every ∊0 > 0, there exists an integer M such that if a permutation π is ∊o-far from in the Kendall's tau distance, then a random subpermutation of π of order M has the property P with probability at most ∊0. This settles an open problem whether hereditary permutation properties are strongly testable, i.e., testable with respect to the Kendall's tau distance, which is considered to be the edit distance for permutations. Our method also yields a proof of a conjecture of Hoppen, Kohayakawa, Moreira and Sampaio on the relation of the rectangular distance and the Kendall's tau distance of a permutation from a hereditary property

Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication

Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions $f_1$, $f_2$ and $f_3: \mathbb{F}_2^k \to \{0, 1\}$ are said to be triangle free if there is no $x, y \in \mathbb{F}_2^k$ such that $f_1(x) = f_2(y) = f_3(x + y) = 1$. This property is known to be strongly testable (Green 2005), but the number of queries needed is upper-bounded only by a tower of twos whose height is polynomial in 1 / \epsislon, where \epsislon is the distance between the tested function triple and triangle-freeness, i.e., the minimum fraction of function values that need to be modified to make the triple triangle free. A lower bound of $(1 / \epsilon)^{2.423}$ for any one-sided tester was given by Bhattacharyya and Xie (2010). In this work we improve this bound to $(1 / \epsilon)^{6.619}$. Interestingly, we prove this by way of a combinatorial construction called \emph{uniquely solvable puzzles} that was at the heart of Coppersmith and Winograd's renowned matrix multiplication algorithm

Maximum flow is approximable by deterministic constant-time algorithm in sparse networks

We show a deterministic constant-time parallel algorithm for finding an almost maximum flow in multisource-multitarget networks with bounded degrees and bounded edge capacities. As a consequence, we show that the value of the maximum flow over the number of nodes is a testable parameter on these networks.Comment: 8 page

Limit structures and property testing

In the thesis, we study properties of large combinatorial objects. We analyze these objects from two different points of view. The first aspect is analytic - we study properties of limit objects of combinatorial structures. We investigate when graphons (limits of graphs) and permutons (limits of permutations) are finitely forcible, i.e., when they are uniquely determined by finitely many densities of their substructures. We give examples of families of permutons that are finitely forcible but the associated graphons are not and we disprove a conjecture of Lovasz and Szegedy on the dimension of the space of typical vertices of a finitely forcible graphon. In particular, we show that there exists a finitely forcible graphon W such that the topological spaces T(W) and T(W) have infinite Lebesgue covering dimension. We also study the dependence between densities of substructures. We prove a permutation analogue of the classical theorem of Erdos, Lovasz and Spencer on the densities of connected subgraphs in large graphs. The second aspect of large combinatorial objects we concentrate on is algorithmic|we study property testing and parameter testing. We show that there exists a bounded testable permutation parameter that is not finitely forcible and that every hereditary permutation property is testable (in constant time) with respect to the Kendall's tau distance, resolving a conjecture of Kohayakawa