In this month’s post, we review papers that raise two questions that I find particularly interesting. First, can sublinear algorithms find applications in efficient approximation of CSPs? And second, can property testing be extended to give meaningful results in settings where Hamming distance is not the “right” measure of distance between functions?

Going for Speed: Sublinear Algorithms for Dense r-CSPs by Grigory Yaroslavtsev (arXiv). In their seminal paper, Goldreich, Goldwasser, and Ron considered the MAX-CUT problem. They showed that both the testing variant of the problem (where the algorithm must distinguish between graphs with a cut of size \(s\) from those with maximum cut size at most \(s – \epsilon N^2\)) and the approximation variant (where the algorithm must find a cut that has at most \(\epsilon N^2\) fewer edges than the largest cut in the graph) can be solved in sublinear time. This result inspired the very successful line of research on testing properties of graphs. We can also view this result as motivation for studying the approximability of general CSPs: given some constraint satisfaction problem, can we find an assignment that satisfies almost as many constraints as the maximally satisfying assignment in sublinear time? This current paper begins the exploration of this question with an improved bound on the query complexity of the MAX-CUT problem and with the first general result for approximating CSPs of arity greater than 2 (i.e., where the constraints are on more than 2 variables).

Lp testing by Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev (preprint). Almost all of the work in testing properties of functions is based on the Hamming distance: for a given property P, a tester is required to accept functions that have the property P and to reject the functions that differ from every function with the property P on an \(\epsilon\) fraction of the inputs. This notion of distance is appropriate for algebraic properties of functions and for properties of boolean functions, but for other properties of real-valued functions, \(L_p\) distances are sometimes more appropriate. This paper provides the first systematic study of property testing under these distances. An important high-level message from this work is that the testing problems can be very different under these distances than under Hamming distance. For example, the classic problem of \(\epsilon\)-testing whether \(f : [n] \to \mathbb{R}\) is monotone requires \(\Theta(\frac{\log n}{\epsilon})\) queries in the Hamming distance setting, but only \(\Theta(\frac1{\epsilon^p})\) queries in the \(L_p\) distance setting!