ITCS 2014 had a nice selection of property testing papers, some of which were covered in our monthly posts. One of our reporters was around, and sent us this report.
Partial tests, universal tests and decomposability by Eldar Fischer, Oded Lachish and Yonatan Goldhirsh. Yonathan talked about the notion of “partial testing”, where one wishes to test a subproperty of a property. This paper has results on when properties cannot be decomposed in testable subproperties, and, as Yonathan hinted at, uses sunflowers! This work is related to recent work of Gur and Rothblum on Merlin-Arthur proofs of proximity.
High Dimensional Expanders and Property Testing by Tali Kaufman and Alexander Lubotsky. Consider a function \(f\) on the vertices of a graph and suppose we wish to check if \(f\) is constant. The simple edge tester queries an edge and checks if both endpoints have distinct values. The success probability of this tester is the expansion of the graph. This paper generalizes this notion to high-dimensional simplicial complices, and this gives a tester for the tensor power property.
Parameterized Testability by Kazuo Iwama and Yuichi Yoshida. Many NP-hard problems are fixed-parameter tractable (FPT). Determining if a graph has a vertex cover of size \(k\) can be done in \(O(poly(n)) + 2^k\) time, as opposed to the trivial \(n^k\) bound. Yuichi talked about getting fixed-parameter testing algorithms, where the tester is constant when the property parameter is constant. The paper has a number of results, notably testing algorithms for \(k\)-vertex feedback set and \(k\)-path freeness.