Computing approximate roots of monotone functions
Alexandros Hollender, Chester Lawrence, Erel Segal-Halevi
TL;DR
The paper addresses the problem of computing $\varepsilon$-roots of Lipschitz vector-valued functions under Miranda's theorem, and identifies monotonicity-based conditions that yield polynomial-time complexity in the accuracy. It shows that in two dimensions a single diagonal or ex-diagonal monotone condition suffices, while in higher dimensions $d^2-d$ ex-diagonal monotonicity is enough to achieve $O(\log^{\lceil (d+1)/2\rceil}(1/\delta))$ evaluations, whereas $d^2-2$ monotonicity conditions can be insufficient. A central contribution is the reduction of envy-free cake-cutting with monotone preferences to an $\varepsilon$-root problem, enabling $r$-near envy-free allocations to be computed with poly-time queries in many group-settings. The results deepen understanding of how monotonicity structure affects root-finding and fixed-point computations, with practical implications for fair division problems and related algorithmic tasks.
Abstract
Given a function f: [a,b] -> R, if f(a) < 0 and f(b)> 0 and f is continuous, the Intermediate Value Theorem implies that f has a root in [a,b]. Moreover, given a value-oracle for f, an approximate root of f can be computed using the bisection method, and the number of required evaluations is polynomial in the number of accuracy digits. The goal of this note is to identify conditions under which this polynomiality result extends to a multi-dimensional function that satisfies the conditions of Miranda's theorem -- the natural multi-dimensional extension of the Intermediate Value Theorem. In general, finding an approximate root might require an exponential number of evaluations even for a two-dimensional function. We show that, if f is two-dimensional and satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. For any fixed dimension d, if f is a d-dimensional function that satisfies all d^2-d ``ex-diagonal'' monotonicity conditions (that is, component i of f is monotonically decreasing with respect to variable j for all i!=j), then the number of required evaluations is polynomial in the accuracy. But if f satisfies only d^2-d-2 ex-diagonal conditions, then the number of required evaluations may be exponential in the accuracy. The case of d^2-d-1 ex-diagonal conditions remains unsolved. As an example application, we show that computing approximate roots of monotone functions can be used for approximate envy-free cake-cutting.