Testing of Matrix Properties

Combinatorial property testing, initiated by Rubinfeld and Sudan [15] and formally defined by Goldreich, Goldwasser and Ron in [12], deals with the following relaxation of decision problems: Given a fixed property P and an input f , distinguish between the case that f satisfies P , and the case that no input that differs from f in less than some fixed fraction of the places satisfies P . An ( ; q)-test for P is a randomized algorithm that queries at most q places of an input x and distinguishes with probability 2/3 between the case that f has the property and the case that at least an -fraction of the places of f need to be changed in order for it to have the property. Here we concentrate on labeled, d-dimensional grids, where the grid is viewed as a partially ordered set (poset) in the standard way (i.e as a product order of total orders). The main result here presents an ( ; poly(1= ))-test for every property of 0/1 labeled, d-dimensional grids that is characterized by a finite collection of forbidden induced posets. Such properties include the `monotonicity' property studied in [7, 6], other more complicated forbidden chain patterns, and general forbidden poset patterns. We also present a (less efficient) test for such properties of labeled grids with larger fixed size alphabets. All the above tests have in addition a 1-sided error probability. Another result is a test for a collection of bipartite graph properties which uses less queries than the previously known algorithms for some of them. Both collections above are variants of properties that are defined by certain first order formulae with no quantifier alternation over the syntax containing the grid order relations (and some additional relations for the bipartite graph properties). We also show that with ..

On the Query Complexity of Testing for Eulerian Orientations

We consider testing directed graphs for being Eulerian in the orientation model introduced in [15]. Despite the local nature of the property of being Eulerian, it turns out to be significantly harder for testing than other properties studied in the orientation model. We show a superconstant lower bound on the query complexity of 2-sided tests and a linear lower bound on the query complexity of 1-sided tests for this property. On the positive side, we give several 1-sided and 2-sided tests, including a sub-linear 2-sided test for general graphs. For special classes of graphs, including bounded-degree graphs and expander graphs, we provide improved results. In particular, for dense graphs we give a 2-sided test with constant query complexity.

A combinatorial characterization of the testable graph properties: it’s all about regularity

A common thread in all the recent results concerning testing dense graphs is the use of Szemerédi’s regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédi-partitions. This means that in some sense, testing for Szemerédi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was first raised in the 1996 paper of Goldreich, Goldwasser and Ron [24] that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable