Statistical exponential formulas for homogeneous diffusion

Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \left\{ \begin{array}{c} u_{t} \ - \ ( \frac{p}{ \, N + p - 2 \, } ) \, \Delta^{1}_{p} u ~ = ~ 0 \quad \mbox{for} \quad x \in \mathbb{R}^{N}, \quad t > 0 \\ \\ u(\cdot,0) ~ = ~ u_{0} \in BUC( \mathbb{R}^{N} ) \end{array} \right. is given by the exponential formula $$u(t) ~ := ~ \lim_{n \to \infty}{ \left( M^{t/n}_{p} \right)^{n} u_{0} } \,$$ where the statistical operator $M^{h}_{p} \colon BUC( \mathbb{R}^{N} ) \to BUC( \mathbb{R}^{N} )$ is defined by $$\left(M^{h}_{p} \varphi \right)(x) := (1-q) \operatorname{median}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, when $1 \leq p \leq 2$ and by $$\left(M^{h}_{p} \varphi \right)(x) := ( 1 - q ) \operatorname{midrange}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } + q \operatorname{mean}_{\partial B(x,\sqrt{2h})}{ \left\{ \, \varphi \, \right\} } \,$$ with $q = \frac{ N }{ N + p - 2 }$, when $p \geq 2$. Possible extensions to problems with Dirichlet boundary conditions and to homogeneous diffusion on metric measure spaces are mentioned briefly

The Unreasonable Success of Local Search: Geometric Optimization

What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude $1/\epsilon^c$ is an approximation scheme for the following problems in the Euclidian plane: TSP with random inputs, Steiner tree with random inputs, facility location (with worst case inputs), and bicriteria $k$-median (also with worst case inputs). The randomness assumption is necessary for TSP

Algoritmos de aproximação para problemas de alocação de instalações e outros problemas de cadeia de fornecimento

Orientadores: Flávio Keidi Miyazawa, Maxim SviridenkoTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O resumo poderá ser visualizado no texto completo da tese digitalAbstract: The abstract is available with the full electronic documentDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã

The Hardness of Approximation of Euclidean k-means

The Euclidean $k$-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of $n$ points in Euclidean space $R^d$, and the goal is to choose $k$ centers in $R^d$ so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general $k$ and a $(1+\epsilon)$-approximation which runs in time $poly(n) 2^{O(k/\epsilon)}$. At the other extreme, the only known computational complexity result for this problem is NP-hardness [ADHP'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in $R^d$ can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all $k,d$. In this paper we provide the first hardness of approximation for the Euclidean $k$-means problem. Concretely, we show that there exists a constant $\epsilon > 0$ such that it is NP-hard to approximate the $k$-means objective to within a factor of $(1+\epsilon)$. We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform $G$, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph $H$, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest