Let A be an arbitrary matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we show that the smoothed precision necessary to solve Ax = b, for any b, using Gaussian elimination without pivoting is logarithmic. Moreover, when A is an all-zero square matrix, our results significantly improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997). Partially supported by NSF grant CCR-0112487

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The probabilistic roadmap algorithm is a leading heuristic for robot motion planning. It is extremely efficient in practice, yet its worst case convergence time is unbounded as a function of the input’s combinatorial complexity. We prove a smoothed polynomial upper bound on the number of samples required to produce an accurate probabilistic roadmap, and thus on the running time of the algorithm, in an environment of simplices. This sheds light on its widespread empirical success. 1

Abstract. We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of ε-tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a spherical disk of radius σ. Besides ε and σ, this bound depends only on the dimension of the sphere and on the degree of the defining equations. 1.

In this paper, we resolve the smoothed and approximative complexity of low-rank quasi-concave minimization, providing both upper and lower bounds. As an upper bound, we provide the first smoothed analysis of quasi-concave minimization. The analysis is based on a smoothed bound for the number of extreme points of the projection of the feasible polytope onto a k-dimensional subspace, where k is the rank (informally, the dimension of nonconvexity) of the quasi-concave function. Our smoothed bound is polynomial in the original dimension of the problem n and the perturbation size ρ, and it is exponential in the rank of the function k. From this, we obtain the first randomized fully polynomialtime approximation scheme for low-rank quasi-concave minimization under broad conditions. In contrast with this, we prove log n-hardness of approximation for general quasi-concave minimization. This shows that our smoothed bound is essentially tight, in that no polynomial smoothed bound is possible for quasi-concave functions of general rank k. The tools that we introduce for the smoothed analysis may be of independent interest. All previous smoothed analyses of polytopes analyzed projections onto two-dimensional subspaces and studied them using trigonometry to examine the angles between vectors and 2-planes in R n. In this paper, we provide what is, to our knowledge, the first smoothed analysis of the projection of polytopes onto higher-dimensional subspaces. To do this, we replace the trigonometry with tools from random matrix theory and differential geometry on the Grassmannian. Our hardness reduction is based on entirely different proofs that may also be of independent interest: we show that the stochastic 2-stage minimum spanning tree problem has a supermodular objective and that su-

Binary search trees are one of the most fundamental data structures. While the height of such a tree may be linear in the worst case, the average height with respect to the uniform distribution is only logarithmic. The exact value is one of the best studied problems in average-case complexity. We investigate what happens in between by analysing the smoothed height of binary search trees: Randomly perturb a given (adversarial) sequence and then take the expected height of the binary search tree generated by the resulting sequence. As perturbation models, we consider partial permutations, partial alterations, and partial deletions. On the one hand, we prove tight lower and upper bounds of roughly Θ((1 − p) · n/p) for the expected height of binary search trees under partial permutations and partial alterations, where n is the number of elements and p is the smoothing parameter. This means that worst-case instances are rare and disappear under slight perturbations. On the other hand, we examine how much a perturbation can increase the height of a binary search tree, i.e. how much worse well balanced instances can become.

We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to ill-posedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every n-by-d matrix Ā, n-vector ¯ b, and d-vector ¯c satisfying ∥ ∥ Ā, ¯ b, ¯c ∥ ∥ F ≤ 1 and every σ ≤ 1, E [log C(A, b, c)] = O(log(nd/σ)), A,b,c where A, b and c are Gaussian perturbations of Ā, ¯ b and ¯c of variance σ 2 and C(A, b, c) is the condition number of the linear program defined by (A, b, c). From this bound, we obtain a smoothed analysis of interior point algorithms. By combining this with the smoothed analysis of finite termination of Spielman and Teng (Math. Prog. Ser. B, 2003), we show that the smoothed complexity of interior point algorithms for linear programming is O(n 3 log(nd/σ)).

The topics of my research in mathematics have varied a lot over time as a PhD student. Beginning in Umeå with extremal graph theory, then combining graph theory with image analysis in order to do image segmentation and finally investigating and improving the modified perceptron algorithm, an algorithm that can be used for segmentation. The results in these different subjects have especially one property in common, that is that they are mainly proved with different discrete methods. Sometimes, as in the case of the geometric properties of the unit sphere in the part concerning the perceptron algorithm, some standard analysis is used and concerning the results about a general perceptron algorithm we get a constructive method of how to achieve a solution to a problem taken from functional analysis. But, the main topics will be discrete, as well in methods used as in the results achieved. Below is a list of my papers that has had importance for the results presented in the thesis. • O. Barr, On extremal graphs without compatible triangles or quadrilaterals, Discrete Mathematics 125 (1994) 31–43. This paper studies a concept of a more general edge colouring, local edge colourings. The results is about how many edges in a graph that will enforce a cycle of length three or four to be compatible with the local edge colouring. • O. Barr, Erdős-Sós Conjecture for Graphs with High Minimum Degree, Research reports, No. 7 (1996), Umeå University, Sweden. This paper studies the well known conjecture by P. Erdős and V. Sós. Here we get an extension of Sidorenkos earlier result and thereby results for graphs with large minimum degree.

The k-means method is a popular algorithm for clustering, known for its speed in practice. This stands in contrast to its exponential worst-case running-time. To explain the speed of the k-means method, a smoothed analysis has been conducted. We sketch this smoothed analysis and a generalization to Bregman divergences. 1 k-Means Clustering The problem of clustering data into classes is ubiquitous in computer science, with applications ranging from computational biology over machine learning to image analysis. The k-means method is a very simple and implementation-friendly local improvement heuristic for clustering. It is used to partition a set X of n d-dimensional data points into k clusters. (The number k of clusters is fixed in advance.) In k-means clustering, our goal is not only to get a clustering of the data points, but also to get a center ci for each cluster Xi of the clustering X1,..., Xk. A center can be viewed as a representative of its cluster. We do not require centers to be among the data points, but they can be arbitrary points. The goal is to find a “good ” clustering, where “good ” means that the clustering should minimize the objective function k ∑ ∑ δ(x, ci). i=1 x∈Xi Here, δ denotes a distance measure. In the following, we will mainly use squared Euclidean distances, i.e., δ(x, ci) = ‖x − ci‖2. Of course, given the cluster centers c1,..., ck ∈ Rd, each point x ∈ X should be assigned to the cluster Xi whose center ci is closest to x. On the other hand, given a clustering X1,..., Xk

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