Parameterized Testability

This article studies property testing for NP optimization problems with parameter k under the general graph model with an augmentation of random edge sampling capability. It is shown that a variety of such problems, including k-Vertex Cover, k-Feedback Vertex Set, k-Multicut, k-Path-Free, and k-Dominating Set, are constant-query testable if k is constant. It should be noted that the first four problems are fixed parameter tractable (FPT) and it turns out that algorithmic techniques for their FPT algorithms (branch-and-bound search, color coding, etc.) are also useful for our testers. k-Dominating Set is W[2]-hard, but we can still test the property with a constant number of queries, since the definition of ε-farness makes the problem trivial for non-sparse graphs that are the source of hardness for the original optimization problem. We also consider k-Odd Cycle Transversal, which is another well-known FPT problem, but we only give a sublinear-query tester when k is a constant.