Directed Isoperimetric Theorems for Boolean Functions on the Hypergrid and an (O)over-tilde (nd) Monotonicity Tester

The problem of testing monotonicity for Boolean functions on the hypergrid, f : [n](d)->{0, 1} is a classic topic in property testing. When n = 2, the domain is the hypercube. For the hypercube case, a breakthrough result of Khot-Minzer-Safra (FOCS 2015) gave a non-adaptive, one-sided tester making (O) over tilde (epsilon(-2)root d) queries. Up to polylog d and epsilon factors, this bound matches the (Omega) over tilde (epsilon root d)-query non-adaptive lower bound (Chen-De-Servedio-Tan (STOC 2015), Chen-Waingarten-Xie (STOC 2017)). For any n > 2, the optimal non-adaptive complexity was unknown. A previous result of the authors achieves a (O) over tilde (d(5/6))-query upper bound (SODA 2020), quite far from the root d bound for the hypercube. In this paper, we resolve the non-adaptive complexity of monotonicity testing for all constant n, up to poly(epsilon(-1) log d) factors. Specifically, we give a non-adaptive, one-sided monotonicity tester making (O) over tilde (epsilon(-2) n root d) queries. From a technical standpoint, we prove newdirected isoperimetric theorems over the hypergrid [n](d). These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube.