An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes

We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from $18$ to $100$ times. We implement our algorithm using the experimental CGAL package {\tt triangulation}. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90\% and 105\% of the true volume, runtime is reduced up to $25$ times. Our method computes instances of $5$-, $6$- or $7$-dimensional polytopes with $35$K, $23$K or $500$ vertices, resp., within $2$hr. Compared to tropical geometry software, ours is faster up to dimension $5$ or $6$, and competitive in higher dimensions

The Gromov Norm of the Product of Two Surfaces

We make an estimation of the value of the Gromov norm of the Cartesian product of two surfaces. Our method uses a connection between these norms and the minimal size of triangulations of the products of two polygons. This allows us to prove that the Gromov norm of this product is between 32 and 52 when both factors have genus 2. The case of arbitrary genera is easy to deduce form this one.Comment: The journal version contains an error that invalidates one direction of the main theorem. The present version contains an erratum, at the end, explaining thi

Plethysm and lattice point counting

We apply lattice point counting methods to compute the multiplicities in the plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition $\mu$ of 3,4, or 5 we obtain an explicit formula in $\lambda$ and $k$ for the multiplicity of $S^\lambda$ in $S^\mu(S^k)$.Comment: 25 pages including appendix, 1 figure, computational results and code available at http://thomas-kahle.de/plethysm.html, v2: various improvements, v3: final version appeared in JFoC

Quantum speedup of classical mixing processes

Most approximation algorithms for #P-complete problems (e.g., evaluating the permanent of a matrix or the volume of a polytope) work by reduction to the problem of approximate sampling from a distribution $\pi$ over a large set $\S$. This problem is solved using the {\em Markov chain Monte Carlo} method: a sparse, reversible Markov chain $P$ on $\S$ with stationary distribution $\pi$ is run to near equilibrium. The running time of this random walk algorithm, the so-called {\em mixing time} of $P$, is $O(\delta^{-1} \log 1/\pi_*)$ as shown by Aldous, where $\delta$ is the spectral gap of $P$ and $\pi_*$ is the minimum value of $\pi$. A natural question is whether a speedup of this classical method to $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$, the diameter of the graph underlying $P$, is possible using {\em quantum walks}. We provide evidence for this possibility using quantum walks that {\em decohere} under repeated randomized measurements. We show: (a) decoherent quantum walks always mix, just like their classical counterparts, (b) the mixing time is a robust quantity, essentially invariant under any smooth form of decoherence, and (c) the mixing time of the decoherent quantum walk on a periodic lattice $\Z_n^d$ is $O(n d \log d)$, which is indeed $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ and is asymptotically no worse than the diameter of $\Z_n^d$ (the obvious lower bound) up to at most a logarithmic factor.Comment: 13 pages; v2 revised several part

Grid graphs, Gorenstein polytopes, and domino stackings

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniques use results from graph theory, polyhedral geometry, and enumerative combinatorics.Comment: 14 pages, 6 figures, uses graphs packag

Gemischte Volumina, gemischte Ehrhart-Theorie und deren Anwendungen in tropischer Geometry und Gestaengekonfigurationsproblemen

The aim of this thesis is the discussion of mixed volumes, their interplay with algebraic geometry, discrete geometry and tropical geometry and their use in applications such as linkage configuration problems. Namely we present new technical tools for mixed volume computation, a novel approach to Ehrhart theory that links mixed volumes with counting integer points in Minkowski sums, new expressions in terms of mixed volumes of combinatorial quantities in tropical geometry and furthermore we employ mixed volume techniques to obtain bounds in certain graph embedding problems.Ziel dieser Arbeit ist die Diskussion gemischter Volumina, ihres Zusammenspiels mit der algebraischen Geometrie, der diskreten Geometrie und der tropischen Geometrie sowie deren Anwendungen im Bereich von Gestaenge-Konfigurationsproblemen. Wir praesentieren insbesondere neue Methoden zur Berechnung gemischter Volumina, einen neuen Zugang zur Ehrhart Theorie, welcher gemischte Volumina mit der Enumeration ganzzahliger Punkte in Minkowski-Summen verbindet, neue Formeln, die kombinatorische Groessen der tropischen Geometrie mithilfe gemischter Volumina beschreiben, und einen neuen Ansatz zur Verwendung gemischter Volumina zur Loesung eines Einbettungsproblems der Graphentheorie

Decomposition Methods for Nonlinear Optimization and Data Mining

We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope into special polyhedra. We use these decompositions and develop methods for computing a special class of integrals exactly. Namely, we are interested in computing the exact value of integrals of polynomial functions over convex polyhedra. We present prior work and new extensions of the integration algorithms. Every integration method we present requires that the polynomial has a special form. We explore two special polynomial decomposition algorithms that are useful for integrating polynomial functions. Both polynomial decompositions have strengths and weaknesses, and we experiment with how to practically use them. After developing practical algorithms and efficient software tools for integrating a polynomial over a polytope, we focus on the problem of maximizing a polynomial function over the continuous domain of a polytope. This maximization problem is NP-hard, but we develop approximation methods that run in polynomial time when the dimension is fixed. Moreover, our algorithm for approximating the maximum of a polynomial over a polytope is related to integrating the polynomial over the polytope. We show how the integration methods can be used for optimization. The second central topic in this dissertation is on problems in data science. We first consider a heuristic for mixed-integer linear optimization. We show how many practical mixed-integer linear have a special substructure containing set partition constraints. We then describe a nice data structure for finding feasible zero-one integer solutions to systems of set partition constraints. Finally, we end with an applied project using data science methods in medical research.Comment: PHD Thesis of Brandon Dutr