Optimization Algorithms for Computational Geometry

Abstract

We study two fundamental problems in computational geometry: finding the maximum inscribed ball (MaxIB) inside a bounded polyhedron defined by \(m\) hyperplanes in a \(d\)-dimensional space, and finding the minimum enclosing ball (MinEB) of a set of \(n\) points in a \(d\)-dimensional space. We translate both these geometric problems into optimization problems and apply first-order methods for smooth and saddle-point optimization to obtain simpler, faster nearly-linear-time algorithms. For MaxIB, the best known running time is \(\tilde{O}(m d \alpha^3 / \varepsilon^3)\) [XSX06], where \(\alpha \geq 1\) is the aspect ratio of the polyhedron. We obtain two new algorithms for MaxIB: one runs in \(\tilde{O}(m d \alpha / \varepsilon)\) time using smooth optimization, and the other runs in \(\tilde{O}(md + m \sqrt{d} \alpha / \varepsilon)\) time using saddle-point optimization. For MinEB, the best known running time is \(\tilde{O}(n d / \sqrt{\varepsilon})\) [SVZ11]. We obtain two new algorithms for MinEB: one runs in \(\tilde{O}(n d / \sqrt{\varepsilon})\) time using smooth optimization, and the other runs in \(\tilde{O}(nd + n \sqrt{d} / \sqrt{\varepsilon})\) time using saddle-point optimization.