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Parameterized Convexity Testing

by   Abhiruk Lahiri, et al.
Chennai Mathematical Institute
University of Haifa
Ariel University

In this work, we develop new insights into the fundamental problem of convexity testing of real-valued functions over the domain [n]. Specifically, we present a nonadaptive algorithm that, given inputs ∈ (0,1), s ∈ℕ, and oracle access to a function, -tests convexity in O(log (s)/), where s is an upper bound on the number of distinct discrete derivatives of the function. We also show that this bound is tight. Since s ≤ n, our query complexity bound is at least as good as that of the optimal convexity tester (Ben Eliezer; ITCS 2019) with complexity O(log n/); our bound is strictly better when s = o(n). The main contribution of our work is to appropriately parameterize the complexity of convexity testing to circumvent the worst-case lower bound (Belovs et al.; SODA 2020) of Ω(log ( n)/) expressed in terms of the input size and obtain a more efficient algorithm.


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