Inflatable graph properties and naturalization of property tests through strong canonicality

We consider natural graph property tests, which act entirely independently of the size of the graph being tested (not just having a number of queries independent of the size). We introduce the notion of graph properties being inflatable — closed under taking (balanced) blowups — and show that the query complexity of natural tests for a property are related to the degree to which it is approximately hereditary and approximately inflatable. Specifically, we show that for properties which are almost hereditary and almost inflatable, any test can be made natural, with a polynomial increase in the number of queries. The naturalization is carried out as a sort of extension of the canonicalization due to Goldreich and Trevisan in [15], so that natural canonical tests can be described as strongly canonical. In the reverse direction, we show that properties admitting natural tests are approximately inflatable and approximately hereditary, with these parameters depending on the test’s number of queries. Using the technique for naturalization, we restore in part the claim in [15] (which was qualified in the errata [16]) regarding testing hereditary properties by ensuring that a small random subgraph itself satisfies the tested property. This restoration allows us to generalize the result of Alon and Shapira in [5], regarding the lower bound on triangle-freeness testing: Any lower bound, not only the currently established quasi-polynomial one, on one-sided testing for triangle freeness holds essentially for two-sided testing as well. We also explore the relations of the notion of inflatability and other already-studied features of properties and property tests in the dense graph model such as one-sidedness, heredity, and proximityoblivion. To Do (not necessarily before finishing my Ph.D. or resubmitting if we don’t get accepted to RANDOM): • Try to salvage the lemma saying that s′ < s natural test for a hereditary-downto-s property implies a one-sided s-test. • Try to construct a hereditary property, which is not 1/10 approximately-inflatable, even with a threshold / on the average. • Try to characterize the properties which are both proximity-oblivious-testable and hereditary (but not necessarily inflatable). • Can we say something interesting about estimation / tolerant testing? ∗Department of Computer Science, Technion, Haifa, Israel. Email: {eldar|eyalroz}@cs.technion.ac.il