Linear programming (LP) relaxations provide a powerful technique to design approximation algorithms for combinatorial optimization problems. In the first part of the thesis, we study the metric s-t path Traveling Salesman Problem (TSP) via LP relaxations. We first consider the s-t path graph-TSP, a critical special case of the metric s-t path TSP. We design a new simple LP-based algorithm for t...

Suppose M is a complete n-dimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T ). For any 0 < p < 1, (p0, t0) ∈ M × (0, T ), q ∈ M , we define the Lp-length between p0 and q, Lp-geodesic, the generalized reduced distance lp and the generalized reduced volume Ṽp(τ), τ = t0 − t, corresponding to the Lp-geodesic at the point p0 at time t0. ...

In this paper, we present a new optimal interpolation error estimate in Lp norm (1 ≤ p ≤ ∞) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We also give new functionals for the global moving mesh method and ...

In this paper, we present a new optimal interpolation error estimate in Lp norm (1 ≤ p ≤ ∞) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We also give new functionals for the global moving mesh method and ...

We study tent spaces on general measure spaces (Ω, μ). We assume that there exists a semigroup of positive operators on Lp(Ω, μ) satisfying a monotone property but do not assume any geometric/metric structure on Ω. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman’s H1-BMO du...

We consider the classical minimum Travelling Salesman Problem on the class of 3-edge-connected cubic graphs. More specifically we consider their (shortest path) metric completions. The well-known conjecture states that the subtour elimination LP relaxation on the min TSP yields a 4/3 approximation factor, yet the best known approximation factor is 3/2. The 3-edge-connected cubic graphs are inte...

Nowak [N], improving a theorem due to A. N. Dranishnikov, G. Gong, V. Lafforgue, and G. Yu [DGLY], gave a characterization of coarse embeddability of general metric spaces into a Hilbert space using a result of Schoenberg on negative definite kernels. He used this characterization to show that the spaces Lp(μ) coarsely embed into a Hilbert space for p < 2. In this article, we show that lp does ...

We discuss two flexibility metrics for Simple Temporal Networks (STNs): the so-called naive flexibility metric based on the difference between earliest and latest starting times of temporal variables, and a recently proposed concurrent flexibility metric. We establish an interesting connection between the computation of these flexibility metrics and properties of the minimal distance matrix DS ...

Let (X, ρ), (Y, σ) be metric spaces and f : X → Y an injective mapping. We put ‖f‖Lip = sup{σ(f(x), f(y))/ρ(x, y); x, y ∈ X, x 6= y}, and dist(f) = ‖f‖Lip .‖f ‖Lip (the distortion of the mapping f). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let X be a finite metric space, and let ε > 0, K be given ...