Computing approximate roots of monotone functions

Alexandros Hollender, Chester Lawrence, Erel Segal-Halevi

Abstract: We are given a value-oracle for a d-dimensional function f that satisfies the conditions of Miranda's theorem, and therefore has a root. Our goal is to compute an approximate root using a number of evaluations that is polynomial in the number of accuracy digits. For d=1<math><mi is="true">d</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">1</mn></math> this is always possible using the bisection method, but for d≥2<math><mi is="true">d</mi><mo is="true">≥</mo><mn is="true">2</mn></math> this is impossible in general.We show that, if d=2<math><mi is="true">d</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn></math> and f satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. The same holds if d≥3<math><mi is="true">d</mi><mo is="true">≥</mo><mn is="true">3</mn></math> and f satisfies some particular d2−d<math><msup is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">d</mi></math> monotonicity conditions. In contrast, if even two of these monotonicity conditions are missing, then the required number of evaluations might be exponential.As an example application, we show that approximate roots of monotone functions can be used for approximate envy-free cake-cutting.