Approximating the distance to properties in bounded-degree and general sparse graphs
ACM Transactions on Algorithms (TALG)
Property Testing: A Learning Theory Perspective
Foundations and Trends® in Machine Learning
Green's conjecture and testing linear-invariant properties
Proceedings of the forty-first annual ACM symposium on Theory of computing
Algorithmic and Analysis Techniques in Property Testing
Foundations and Trends® in Theoretical Computer Science
Approximating the distance to monotonicity in high dimensions
ACM Transactions on Algorithms (TALG)
Parameter testing in bounded degree graphs of subexponential growth
Random Structures & Algorithms
Local Monotonicity Reconstruction
SIAM Journal on Computing
Approximate Hypergraph Partitioning and Applications
SIAM Journal on Computing
Inflatable graph properties and natural property tests
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Testing Eulerianity and connectivity in directed sparse graphs
Theoretical Computer Science
SIAM Journal on Discrete Mathematics
Every locally characterized affine-invariant property is testable
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Testing linear-invariant function isomorphism
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
| Hi-index | 0.00 |
Tolerant testing is an emerging topic in the field of property testing, which was defined in [M. Parnas, D. Ron, and R. Rubinfeld, J. Comput. System Sci., 72 (2006), pp. 1012-1042] and has recently become a very active topic of research. In the general setting, there exist properties that are testable but are not tolerantly testable [E. Fischer and L. Fortnow, Proceedings of the $20$th IEEE Conference on Computational Complexity, 2005, pp. 135-140]. On the other hand, we show here that in the setting of the dense graph model, all testable properties are not only tolerantly testable (which was already implicitly proved in [N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy, Combinatorica, 20 (2000), pp. 451-476] and [O. Goldreich and L. Trevisan, Random Structures Algorithms, 23 (2003), pp. 23-57]), but also admit a constant query size algorithm that estimates the distance from the property up to any fixed additive constant. In the course of the proof we develop a framework for extending Szemerédi's regularity lemma, both as a prerequisite for formulating what kind of information about the input graph will provide us with the correct estimation, and as the means for efficiently gathering this information. In particular, we construct a probabilistic algorithm that finds the parameters of a regular partition of an input graph using a constant number of queries, and an algorithm to find a regular partition of a graph using a $\mathrm{TC}_0$ circuit. This, in some ways, strengthens the results of [N. Alon, R. A. Duke, H. Lefmann, V. Rödl, and R. Yuster, J. Algorithms, 16 (1994), pp. 80-109].