Tobias Brunsch

Lower Bounds for the Average and Smoothed Number of Pareto-Optima

Smoothed analysis of multiobjective 0-1 linear optimization has drawn con- siderable attention recently. The goal is to give bounds for the number of Pareto-optimal solutions (i. e., solutions with the property that no other solution is at least as good in all the coordinates and better in at least one) for multiobjective optimization problems. In this article we prove several lower bounds for the expected number of Pareto optima. Our basic result is a lower bound of Wd(n d 1 ) for optimization problems with d objectives and n variables under fairly general conditions on the distributions of the linear objectives. Our proof relates the problem of finding lower bounds for the number of Pareto optima to results in discrete geometry and geometric probability about arrangements of hyperplanes. We use our basic result to derive the following results: (1) To our knowledge, the first lower bound for natural multiobjective optimization problems. We illustrate this on the maximum spanning tree problem with randomly chosen edge weights. Our technique is sufficiently flexible to yield such lower bounds also for other standard objective functions studied in this setting (such as multiobjective shortest path, TSP, matching). (2) A smoothed lower bound of minfWd(n d 1:5 f d ); 2 Qd(n) g for f -smooth instances of the 0-1 knapsack problem with d profits.

Smoothed Analysis of Belief Propagation for Minimum-Cost Flow and Matching ?

Belief propagation (BP) is a message-passing heuristic for statistical inference in graphical models such as Bayesian networks and Markov random fields. BP is used to compute marginal distributions or maximum likelihood assignments and has applications in many areas, including machine learning, image processing, and computer vision. However, the theoretical understanding of the performance of BP is unsatisfactory. Recently, BP has been applied to combinatorial optimization problems. It has been proved that BP can be used to compute maximum-weight matchings and minimum-cost flows for instances with a unique optimum. The number of iterations needed for this is pseudo-polynomial and hence BP is not efficient in general. We study belief propagation in the framework of smoothed analysis and prove that with high probability the number of iterations needed to compute maximum-weight matchings and minimum-cost flows is bounded by a polynomial if the weights/costs of the edges are randomly perturbed. To prove our upper bounds, we use an isolation lemma by Beier and Vocking (SIAM J. Comput., 2006) for matching and generalize an isolation lemma for min-cost flow by Gamarnik, Shah, and Wei (Oper. Res., 2012). We also prove almost matching lower tail bounds for the number of iterations that BP needs to converge.

Finding short paths on polytopes by the shadow vertex algorithm

We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope

A bad instance for k-means++

P = \left\{ x \in \mathbb{R}^n \,\colon\, Ax \leq b \right\}

Improved Smoothed Analysis of Multiobjective Optimization

along the edges of P, where A∈ℝm ×n. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/δ that is a measure for the flatness of the vertices of P. For integer matrices A∈ℤm ×n we show a connection between δ and the largest absolute value Δ of any sub-determinant of A, yielding a bound of O(Δ4mn4) for the length of the computed path. This bound is expressed in the same parameter Δ as the recent non-constructive bound of O(Δ2n4 log(n Δ)) by Bonifas et al. [1]. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(mn4), which significantly improves the previously best known constructive bound of O(m16n3 log3 (mn)) by Dyer and Frieze [7].

Solving Totally Unimodular LPs with the Shadow Vertex Algorithm

k-means++ is a seeding technique for the k-means method with an expected approximation ratio of O(log k), where k denotes the number of clusters. Examples are known on which the expected approximation ratio of k-means++ is Ω(log k), showing that the upper bound is asymptotically tight. However, it remained open whether k-means++ yields an O(1)-approximation with probability 1/poly(k) or even with constant probability. We settle this question and present instances on which k-means++ achieves an approximation ratio of (2/3-e) ċ log k only with exponentially small probability.

Bounds for the Convergence Time of Local Search in Scheduling Problems

We present several new results about smoothed analysis of multiobjective optimization problems. Motivated by the discrepancy between worst-case analysis and practical experience, this line of research has gained a lot of attention in the last decade. We consider problems in which d linear and one arbitrary objective function are to be optimized over a set S ⊆ {0, 1}n of feasible solutions. We improve the previously best known bound for the smoothed number of Pareto-optimal solutions to O(n2d φd), where φ denotes the perturbation parameter. Additionally, we show that for any constant c the cth moment of the smoothed number of Pareto-optimal solutions is bounded by O((n2d φd)c). This improves the previously best known bounds significantly. Furthermore, we address the criticism that the perturbations in smoothed analysis destroy the zero-structure of problems by showing that the smoothed number of Pareto-optimal solutions remains polynomially bounded even for zero-preserving perturbations. This broadens the class of problems captured by smoothed analysis and it has consequences for nonlinear objective functions. One corollary of our result is that the smoothed number of Pareto-optimal solutions is polynomially bounded for polynomial objective functions. Our results also extend to integer optimization problems.

Improved smoothed analysis of multiobjective optimization

We show that the shadow vertex simplex algorithm can be used to solve linear programs in strongly polynomial time with respect to the number

Smoothed performance guarantees for local search

n

Lower bounds for the smoothed number of pareto optimal solutions

of variables, the number