In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

Journal of the ACM

2004아테네 올림픽대회출전 한국레슬링 대표선수의 무산소성 운동능력 및 등속성 근력

여성사 쓰기의 가능성과 현주소, 한국여성연구소 여성사연구실 지음, 청년사 1999

【臨床医学の展望 2012】 呼吸器病学

Let G be a graph with n vertices and maximum degree d. Fix some minor-closed property P (such as planarity). We say that G is e-far from P if one has to remove e dn edges to make it have P. The problem of property testing P was introduced in the seminal work of Benjamini-Schramm-Shapira (STOC 2008) that gave a tester with query complexity triply exponential in e−1. Levi-Ron (TALG 2015) have given the best tester to date, with a quasipolynomial (in e−1) query complexity. It is an open problem to get property testers whose query complexity is (de−1), even for planarity. In this paper, we resolve this open question. For any minor-closed property, we give a tester with query complexity d· (e−1). The previous line of work on (independent of n, two-sided) testers is primarily combinatorial. Our work, on the other hand, employs techniques from spectral graph theory. This paper is a continuation of recent work of the authors (FOCS 2018) analyzing random walk algorithms that find forbidden minors.

https://par.nsf.gov/servlets/purl/10110941

Random walks and forbidden minors II: a poly(d ε-1)-query tester for minor-closed properties of bounded degree graphs

Let $G$ be an undirected, bounded degree graph with $n$ vertices. Fix a finite graph $H$, and suppose one must remove $\varepsilon n$ edges from $G$ to make it $H$-minor free (for some small constant $\varepsilon > 0$). We give an $n^{1/2+o(1)}$-time randomized procedure that, with high probability, finds an $H$-minor in such a graph. As an application, suppose one must remove $\varepsilon n$ edges from a bounded degree graph $G$ to make it planar. This result implies an algorithm, with the same running time, that produces a $K_{3,3}$ or $K_5$ minor in $G$. No prior sublinear time bound was known for this problem. 
By the graph minor theorem, we get an analogous result for any minor-closed property. Up to $n^{o(1)}$ factors, this resolves a conjecture of Benjamini-Schramm-Shapira (STOC 2008) on the existence of one-sided property testers for minor-closed properties. Furthermore, our algorithm is nearly optimal, by an $\Omega(\sqrt{n})$ lower bound of Czumaj et al (RSA 2014). 
Prior to this work, the only graphs $H$ for which non-trivial one-sided property testers were known for $H$-minor freeness are the following: $H$ being a forest or a cycle (Czumaj et al, RSA 2014), $K_{2,k}$, $(k\times 2)$-grid, and the $k$-circus (Fichtenberger et al, Arxiv 2017).

Finding forbidden minors in sublinear time: a $n^{1/2+o(1)}$-query one-sided tester for minor closed properties on bounded degree graphs

A Boolean $k$-monotone function defined over a finite poset domain ${\cal D}$ alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in ${\cal D}$. Therefore, $k$-monotone functions are natural generalizations of the classical monotone functions, which are the $1$-monotone functions. Motivated by the recent interest in $k$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $k$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $k$-monotone (or are close to being $k$-monotone) from functions that are far from being $k$-monotone. Our results include the following: 
- We demonstrate a separation between testing $k$-monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$; 
- We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=\omega(\log d)$: testing $k$-monotonicity can be performed with $2^{O(\sqrt d \cdot \log d\cdot \log{1/\varepsilon})}$ queries, while learning $k$-monotone functions requires $2^{\Omega(k\cdot \sqrt d\cdot{1/\varepsilon})}$ queries (Blais et al. (RANDOM 2015)). 
- We present a tolerant test for functions $f\colon[n]^d\to \{0,1\}$ with complexity independent of $n$, which makes progress on a problem left open by Berman et al. (STOC 2014). 
Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid $[n]^d$, and draw connections to distribution testing techniques.

Testing $k$-Monotonicity

A function $f:\{0,1\}^n\rightarrow \{0,1\}$ is said to be $k$-monotone if it flips between 0 and 1 at most $k$ times on every ascending chain. Such functions represent a natural generalization of (...

Akash Kumar

Papers

Random walks and forbidden minors II: a poly(d ε-1)-query tester for minor-closed properties of bounded degree graphs

Finding forbidden minors in sublinear time: a $n^{1/2+o(1)}$-query one-sided tester for minor closed properties on bounded degree graphs

Testing $k$-Monotonicity

Flipping out with Many Flips: Hardness of Testing k-Monotonicity.

Testing k-Monotonicity.