Abstract
We present a general notion of properties that are characterized by local conditions that are invariant under a sufficiently rich class of symmetries. Our framework generalizes two popular models of testing graph properties as well as the algebraic invariances studied by Kaufman and Sudan (STOC’08). Our focus is on the case that the property is characterized by a constant number of local conditions and a rich set of invariances.
We show that, in the aforementioned models of testing graph properties, characterization by such invariant local conditions is closely related to proximity oblivious testing (as defined by Goldreich and Ron, STOC’09). In contrast to this relation, we show that, in general, characterization by invariant local conditions is neither necessary nor sufficient for proximity oblivious testing. Furthermore, we show that easy testability is not guaranteed even when the property is characterized by local conditions that are invariant under a 1-transitive group of permutations.
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Goldreich, O., Kaufman, T. (2011). Proximity Oblivious Testing and the Role of Invariances. In: Goldreich, O. (eds) Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation. Lecture Notes in Computer Science, vol 6650. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22670-0_19
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DOI: https://doi.org/10.1007/978-3-642-22670-0_19
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