Proofs of Proximity for Distribution Testing

Distribution testing is an area of property testing that studies algorithms that receive few samples from a probability distribution D and decide whether D has a certain property or is far (in total variation distance) from all distributions with that property. Most natural properties of distributions, however, require a large number of samples to test, which motivates the question of whether there are natural settings wherein fewer samples suffice. We initiate a study of proofs of proximity for properties of distributions. In their basic form, these proof systems consist of a tester that not only has sample access to a distribution but also explicit access to a proof string that depends on the distribution. We refer to these as NP distribution testers, or MA distribution testers if the tester is a probabilistic algorithm. We also study the more general notion of IP distribution testers, in which the tester interacts with an all-powerful untrusted prover. We investigate the power and limitations of proofs of proximity for distributions and chart a landscape that, surprisingly, is significantly different from that of proofs of proximity for functions. Our main results include showing that MA distribution testers can be quadratically stronger than standard distribution testers, but no stronger than that; in contrast, IP distribution testers can be exponentially stronger than standard distribution testers, but when restricted to public coins they can be at best quadratically stronger

An exponential separation between MA and AM proofs of proximity

Interactive proofs of proximity allow a sublinear-time verifier to check that a given input is close to the language, using a small amount of communication with a powerful (but untrusted) prover. In this work we consider two natural minimally interactive variants of such proofs systems, in which the prover only sends a single message, referred to as the proof. The first variant, known as MA-proofs of Proximity (MAP), is fully non-interactive, meaning that the proof is a function of the input only. The second variant, known as AM-proofs of Proximity (AMP), allows the proof to additionally depend on the verifier's (entire) random string. The complexity of both MAPs and AMPs is the total number of bits that the verifier observes - namely, the sum of the proof length and query complexity. Our main result is an exponential separation between the power of MAPs and AMPs. Specifically, we exhibit an explicit and natural property Pi that admits an AMP with complexity O(log n), whereas any MAP for Pi has complexity Omega~(n^{1/4}), where n denotes the length of the input in bits. Our MAP lower bound also yields an alternate proof, which is more general and arguably much simpler, for a recent result of Fischer et al. (ITCS, 2014). Lastly, we also consider the notion of oblivious proofs of proximity, in which the verifier's queries are oblivious to the proof. In this setting we show that AMPs can only be quadratically stronger than MAPs. As an application of this result, we show an exponential separation between the power of public and private coin for oblivious interactive proofs of proximity

Finding Cycles and Trees in Sublinear Time

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least $k\geq 3$ and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being $C_k$-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., $\Omega(1)$-far) {from} being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time \tildeO(\sqrt{N}), where $N$ denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors. The foregoing results are the outcome of our study of the complexity of {\em one-sided error} property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of $N$-vertex graphs can be tested with one-sided error within time complexity \tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known $\Omega(\sqrt{N})$ query lower bound, and contrasts with the fact that any minor-free property admits a {\em two-sided error} tester of query complexity that only depends on the proximity parameter \e. For any constant $k\geq3$, we extend this result to testing whether the input graph has a simple cycle of length at least $k$. On the other hand, for any fixed tree $T$, we show that $T$-minor-freeness has a one-sided error tester of query complexity that only depends on the proximity parameter \e. Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in $o(\sqrt{N})$ complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree Graphs, One-Sided vs Two-Sided Error Probability Updated versio

Succinct Interactive Oracle Proofs: Applications and Limitations

\textit{Interactive Oracle Proofs} (IOPs) are a new type of proof-system that combines key properties of interactive proofs and PCPs: IOPs enable a verifier to be convinced of the correctness of a statement by interacting with an untrusted prover while reading just a few bits of the messages sent by the prover. IOPs have become very prominent in the design of efficient proof-systems in recent years. In this work we study \textit{succinct IOPs}, which are IOPs in which the communication complexity is polynomial (or even linear) in the original witness. While there are strong impossibility results for the existence of succinct PCPs (i.e., PCPs whose length is polynomial in the witness), it is known that the rich class of NP relations that are decidable in small space have succinct IOPs. In this work we show both new applications, and limitations, for succinct IOPs: \begin{itemize} \item First, using one-way functions, we show how to compile IOPs into zero-knowledge \textit{proofs}, while nearly preserving the proof length. This complements a recent line of work, initiated by Ben~Sasson~\etal{}~(TCC, 2016B), who compile IOPs into super-succinct zero-knowledge \textit{arguments}. Applying the compiler to the state-of-the-art succinct IOPs yields zero-knowledge proofs for bounded-space NP relations, with communication that is nearly equal to the original witness length. This yields the shortest known zero-knowledge proofs from the minimal assumption of one-way functions. \item Second, we give a barrier for obtaining succinct IOPs for more general NP relations. In particular, we show that if a language has a succinct IOP, then it can be decided in \textit{space} that is proportionate only to the witness length, after a bounded-time probabilistic preprocessing. We use this result to show that under a simple and plausible (but to the best of our knowledge, new) complexity-theoretic conjecture, there is no succinct IOP for CSAT. \end{itemize