Stephen A. Vavasis

Quadratic programming with one negative eigenvalue is NP-hard

We show that the problem of minimizing a concave quadratic function with one concave direction is NP-hard. This result can be interpreted as an attempt to understand exactly what makes nonconvex quadratic programming problems hard. Sahni in 1974 [8] showed that quadratic programming with a negative definite quadratic term (n negative eigenvalues) is NP-hard, whereas Kozlov, Tarasov and Hačijan [2] showed in 1979 that the ellipsoid algorithm solves the convex quadratic problem (no negative eigenvalues) in polynomial time. This report shows that even one negative eigenvalue makes the problem NP-hard.

On the Complexity of Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) has become a prominent technique for the analysis of image databases, text databases, and other information retrieval and clustering applications. The problem is most naturally posed as continuous optimization. In this report, we define an exact version of NMF. Then we establish several results about exact NMF: (i) that it is equivalent to a problem in polyhedral combinatorics; (ii) that it is NP-hard; and (iii) that a polynomial-time local search heuristic exists.

Nonlinear optimization: complexity issues

Optimization and convexity complexity theory convex quadratic programming non-convex quadratic programming local optimization complexity in the black-box model.

Separators for sphere-packings and nearest neighbor graphs

A collection of <italic>n</italic> balls in <italic>d</italic> dimensions forms a <italic>k</italic>-ply system if no point in the space is covered by more than <italic>k</italic> balls. We show that for every <italic>k</italic>-ply system Γ, there is a sphere <italic>S</italic> that intersects at most <italic>O</italic>(<italic>k</italic><supscrpt>1/<italic>d</italic></supscrpt><italic>n</italic><supscrpt>1−1/<italic>d</italic></supscrpt>) balls of Γ and divides the remainder of Γ into two parts: those in the interior and those in the exterior of the sphere <italic>S</italic>, respectively, so that the larger part contains at most (1−1/(<italic>d</italic>+2))<italic>n</italic> balls. This bound of (<italic>O</italic>(<italic>k</italic><supscrpt>1/<italic>d</italic></supscrpt><italic>n</italic><supscrpt>1−1/<italic>d</italic></supscrpt>) is the best possible in both <italic>n</italic> and <italic>k</italic>. We also present a simple randomized algorithm to find such a sphere in <italic>O(n)</italic> time. Our result implies that every <italic>k</italic>-nearest neighbor graphs of <italic>n</italic> points in <italic>d</italic> dimensions has a separator of size <italic>O</italic>(<italic>k</italic><supscrpt>1/<italic>d</italic></supscrpt><italic>n</italic><supscrpt>1−1/<italic>d</italic></supscrpt>). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.

A unified geometric approach to graph separators

A class of graphs called k-overlap graphs is proposed. Special cases of k-overlap graphs include planar graphs, k-nearest neighbor graphs, and earlier classes of graphs associated with finite element methods. A separator bound is proved for k-overlap graphs embedded in d dimensions. The result unifies several earlier separator results. All the arguments are based on geometric properties of embedding. The separator bounds come with randomized linear-time and randomized NC algorithms. Moreover, the bounds are the best possible up to the leading term.<<ETX>>

Fast and Robust Recursive Algorithmsfor Separable Nonnegative Matrix Factorization

In this paper, we study the nonnegative matrix factorization problem under the separability assumption (that is, there exists a cone spanned by a small subset of the columns of the input nonnegative data matrix containing all columns), which is equivalent to the hyperspectral unmixing problem under the linear mixing model and the pure-pixel assumption. We present a family of fast recursive algorithms and prove they are robust under any small perturbations of the input data matrix. This family generalizes several existing hyperspectral unmixing algorithms and hence provides for the first time a theoretical justification of their better practical performance.This paper describes the implementation and performance of SPRINT, an interactive system for printed circuit board design developed at the Stanford Linear Accelerator Center (SLAC). Topics discussed include the placement subsystem, the routing subsystem consisting of an interactive manual router, an automatic batch router, and a via elimination program, as well as the structure of the design file around which the entire system is centered.

Automatic Mesh Partitioning

This paper describes an efficient approach to partitioning unstructured meshes that occur naturally in the finite element and finite difference methods. The approach makes use of the underlying geometric structure of a given mesh and finds a provably good partition in random O(n) time. It applies to meshes in both two and three dimensions. The new method has applications in efficient sequential and parallel algorithms for large-scale problems in scientific computing. This is an overview paper written with emphasis on the algorithmic aspects of the approach. Many detailed proofs can be found in companion papers.

Quality mesh generation in three dimensions

We show how to triangulate a three dimensional polyhedral region with holes. Our triangulation is optimal in the following two senses. First, our triangulation achieves the best possible aspect ratio up to a constant. Second, for any other triangulation of the same region into m triangles with bounded aspect ratio, our triangulation has size n = O(m). Such a triangulation is desired as an initial mesh for a finite element mesh refinement algorithm. Previous three dimensional triangulation schemes either worked only on a restricted class of input, or did not guarantee well-shaped tetrahedra, or were not able to bound the output size. We build on some of the ideas presented in previous work by Bern, Eppstein, and Gilbert, who have shown how to triangulate a two dimensional polyhedral region with holes, with similar quality and optimality bounds.

Preconditioning for boundary integral equations

New classes of preconditioners are proposed for the linear systems arising from a boundary integral equation method. The problem under consideration is Laplace’s equation in three dimensions. The s...

Geometric Separators for Finite-Element Meshes

We propose a class of graphs that would occur naturally in finite-element and finite-difference problems and we prove a bound on separators for this class of graphs. Graphs in this class are embedded in