We’ve got a nice collection of papers this August, ranging from graph property testing, locally testable codes, to new lower bound approaches.
Deconstructions of Reductions from Communication Complexity to Property Testing using Generalized Parity Decision Trees by Roei Tell (ECCC). Blais, Brody, and Matulef (BBM) introduced the beautiful and powerful method of using communication complexity to derive property testing lower bounds. This result is all about understanding this approach is greater depth, and really trying to get at the crux of the reductions. The BBM approach uses rather heavy hammers (communication complexity lower bounds), so could we get away with weaker hammers to get property testing lower bounds? The key insight is that less powerful models can be reduced to existing property testing problems, and these are equivalent to the BBM reductions. While this (as of now) does not lead to newer bounds, it does shed significant light on existing bounds. The weaker model is essentially that of linear decision trees. Such reductions were also discovered by Brushundi, Chakraborty, and Kulkarni, proving \(\Omega(k)\) lower bounds for testing \(k\)-linearity.
Locally Correctable and Testable Codes Approaching the Singleton Bound by Or Meir (ECCC). Locally Testable Codes (LTCs) are codes that have a constant (or polylogarithmic) time tester. It is known that such codes with minor caveats cannot have a constant rate (a number of papers on this topic: Katz and Trevisan, Dinur and Kaufman, Ben-Sasson and Viderman). But when the query complexity can be \(n^\beta\) (where \(n\) is the codeword size), the rate can be arbitrarily close to \(1\) (Viderman, and more refs in the paper). This result takes this fact to the very extreme. The Singleton Bound is a lower bound between the tradeoff between rate and relative distance of a code. Amazingly, this lower bound tradeoff can be achieved by an LTC (and also Locally Correctable Code) testable with \(n^\beta\) queries!
Complexity of Nondeterministic Graph Parameter Testing by Marek Karpinksi and Ronald Markó (arXiv). It is easiest to think of nondeterminstic graph testing with the example of testing bipartiteness for dense graphs. Suppose we wish to test \(G\) for bipartiteness. As a “certificate”, we are provided a candidate partition \(V = V_1 \cup V_2\). There is a “simple” constant-time tester that can property testing if \(G\) is bipartite with respect to this partition. (Just sample two uniform random vertices, and check if the edge between them respects the bipartition.) So we say that bipartiteness is non-determinstically testable: if \(G\) is far from being bipartite, no certificate can convince the simple tester. The full definition is somewhat complex, but it is basically the same idea. (In this context, “testablility” always refers to constant time testable.) Lovász and Vestergombi proved the striking result that any non-deterministically testable property is also deterministically testable, but their proof did not give an explicit bound on the query complexity. Gishboliner and Shapira fixed that, and gave a tower-type bound for the query complexity. So if the query complexity of the underlying “simple” tester is \(q\), then the non-deterministic property has a tester whose query complexity is a tower of height \(q\). This results goes much further, and shows the final query complexity is at most triply exponential in \(q\). This is quite significant, since it is uses a weaker regularity approach than previous results.