Boolean Unateness Testing with Õ(n^{3/4}) Adaptive Queries

Abstract

We give an adaptive algorithm which tests whether an unknown Boolean function f : {0, 1} n → {0, 1} is unate, i.e. every variable of f is either non-decreasing or non-increasing, or ε-far from unate with one-sided error using O(n 3/4 /ε 2 ) queries. This improves on the best adaptive O(n/ε)-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri [BCP + 17b] when 1/ε n 1/4 . Combined with the Ω(n)-query lower bound for non-adaptive algorithms with one-sided error of [CWX17, BCP + 17a], we conclude that adaptivity helps for the testing of unateness with one-sided error. A crucial component of our algorithm is a new subroutine for finding bi-chromatic edges in the Boolean hypercube called adaptive edge search.

Testing unateness nearly optimally

“…We note that our Scores Lemma above looks very similar to Lemma 4.3 from [CWX17b]. Thus the proof follows a similar trajectory.…”

“…The fact that at least half of edges in G are -strong follows directly from the next two claims: The proof of Claim 4.8 follows from the arguments in Section 6.2 in [CWX17b]. Specifically, given the definition of robust sets for a bichromatic edge e of a certain size in Definition 6.4 of [CWX17b], Claim 4.8 is equivalent to applying Lemma 6.11 and Lemma 6.12 twice.…”

“…Indeed, if (2) held for S of size larger than √ n, then one could improve on the O( √ n)-query algorithm of [KMS15] for testing monotonicity. Consequently, if one believes that monotonicity testing requires Ω( √ n) adaptive queries, it is natural to conjecture that the algorithm in [CWX17b] is optimal for testing unateness.…”

“…The key insight in this work is to preprocess the set S before using AE-Search. For our simplified setting, we first sample S 0 ⊂ [n] of size n 2/3 (much larger than what the analysis in [KMS15,CWX17b,CS18] would allow) uniformly at random. Then, we set S = S 0 , and repeat the following steps for n 2/3 · polylog(n) many iterations:…”

“…(3) The set I can be much smaller than n, in which case, the techniques from [CWX17b] actually achieves better query complexity. We formalize this in Case 3 of the algorithm.…”