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Consider property testing on bounded degree graphs and let $\varepsilon>0$ denote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are…
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…
We study quantum property testing for directed graphs with maximum in-degree and out-degree bounded by some universal constant $d$. For a proximity parameter $\varepsilon$, we show that any property that can be tested with…
We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether an $ n $-vertex graph contains a triangle, clique or star of some size. For a general subgraph $ H $ with $ k $…
Graph-based tests are a class of non-parametric two-sample tests useful for analyzing high-dimensional data. The test statistics are constructed from similarity graphs (such as K-minimum spanning tree), and consequently, their performance…
An assignment of colours to the vertices of a graph is stable if any two vertices of the same colour have identically coloured neighbourhoods. The goal of colour refinement is to find a stable colouring that uses a minimum number of…
Property testing algorithms are highly efficient algorithms, that come with probabilistic accuracy guarantees. For a property P, the goal is to distinguish inputs that have P from those that are far from having P with high probability…
In order to make more complex number-based strings from topological coding for defending against the intelligent attacks equipped with quantum computing and providing effective protection technology for the age of quantum computing, we will…
We present several new examples of speed-ups obtainable by quantum algorithms in the context of property testing. First, motivated by sampling algorithms, we consider probability distributions given in the form of an oracle $f:[n]\to[m]$.…
The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are $\epsilon$-far from satisfying the property. There are now several general results in this area which show that natural…
The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph $H$ is present in an input graph $G$, has been the subject of considerable study. However, even for relatively simple subgraphs, such as paths…
We provide a combinatorial characterization of all testable properties of $k$-uniform hypergraphs ($k$-graphs for short). Here, a $k$-graph property $P$ is testable if there is a randomized algorithm which makes a bounded number of edge…
We study the two inference problems of detecting and recovering an isolated community of \emph{general} structure planted in a random graph. The detection problem is formalized as a hypothesis testing problem, where under the null…
Symmetries of combinatorial objects are known to complicate search algorithms, but such obstacles can often be removed by detecting symmetries early and discarding symmetric subproblems. Canonical labeling of combinatorial objects…
We study the problem of efficiently certifying upper bounds on the independence number of $\ell$-uniform hypergraphs. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within…
The isomorphism problem is a fundamental problem in network analysis, which involves capturing both low-order and high-order structural information. In terms of extracting low-order structural information, graph isomorphism algorithms…
The area of graph property testing seeks to understand the relation between the global properties of a graph and its local statistics. In the classical model, the local statistics of a graph is defined relative to a uniform distribution…
We give an extension of a graph result by Alon and Shapira. And it affirmatively settles a question on property testing raised by them. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our…
We introduce the {\em certification} of solutions to graph problems when access to the input is restricted. This topic has received a lot of attention in the distributed computing setting, and we introduce it here in the context of…
We prove that the combinatorial Weisfeiler-Leman algorithm of dimension $(3k+4)$ is a complete isomorphism test for the class of all graphs of rank width at most $k$. Rank width is a graph invariant that, similarly to tree width, measures…