Solving Totally Unimodular LPs with the Shadow Vertex Algorithm

We show that the shadow vertex simplex algorithm can be used to solve linear programs in strongly polynomial time with respect to the number n of variables, the number m of constraints, and 1/delta, where delta is a parameter that measures the flatness of the vertices of the polyhedron. This extends our recent result that the shadow vertex algorithm finds paths of polynomial length (w.r.t. n, m, and 1/delta) between two given vertices of a polyhedron [4]. Our result also complements a recent result due to Eisenbrand and Vempala [6] who have shown that a certain version of the random edge pivot rule solves linear programs with a running time that is strongly polynomial in the number of variables n and 1/delta, but independent of the number m of constraints. Even though the running time of our algorithm depends on m, it is significantly faster for the important special case of totally unimodular linear programs, for which 1/deltale n and which have only O(n^2) constraints

Smoothed Analysis of Selected Optimization Problems and Algorithms

Optimization problems arise in almost every field of economics, engineering, and science. Many of these problems are well-understood in theory and sophisticated algorithms exist to solve them efficiently in practice. Unfortunately, in many cases the theoretically most efficient algorithms perform poorly in practice. On the other hand, some algorithms are much faster than theory predicts. This discrepancy is a consequence of the pessimism inherent in the framework of worst-case analysis, the predominant analysis concept in theoretical computer science. We study selected optimization problems and algorithms in the framework of smoothed analysis in order to narrow the gap between theory and practice. In smoothed analysis, an adversary specifies the input, which is subsequently slightly perturbed at random. As one example we consider the successive shortest path algorithm for the minimumcost flow problem. While in the worst case the successive shortest path algorithm takes exponentially many steps to compute a minimum-cost flow, we show that its running time is polynomial in the smoothed setting. Another problem studied in this thesis is makespan minimization for scheduling with related machines. It seems to be unlikely that there exist fast algorithms to solve this problem exactly. This is why we consider three approximation algorithms: the jump algorithm, the lex-jump algorithm, and the list scheduling algorithm. In the worst case, the approximation guarantees of these algorithms depend on the number of machines. We show that there is no such dependence in smoothed analysis. We also apply smoothed analysis to multicriteria optimization problems. In particular, we consider integer optimization problems with several linear objectives that have to be simultaneously minimized. We derive a polynomial upper bound for the size of the set of Pareto-optimal solutions contrasting the exponential worst-case lower bound. As the icing on the cake we find that the insights gained from our smoothed analysis of the running time of the successive shortest path algorithm lead to the design of a randomized algorithm for finding short paths between two given vertices of a polyhedron. We see this result as an indication that, in future, smoothed analysis might also result in the development of fast algorithms.Optimierungsprobleme treten in allen wirtschaftlichen, naturwissenschaftlichen und technischen Gebieten auf. Viele dieser Probleme sind ausführlich untersucht und aus praktischer Sicht effizient lösbar. Leider erweisen sich in vielen Fällen die theoretisch effizientesten Algorithmen in der Praxis als ungeeignet. Auf der anderen Seite sind einige Algorithmen viel schneller als die Theorie vorhersagt. Dieser scheinbare Widerspruch resultiert aus dem Pessimismus, der dem in der theoretischen Informatik vorherrschenden Analysekonzept, der Worst-Case-Analyse, innewohnt. Um die Lücke zwischen Theorie und Praxis zu verkleinern, untersuchen wir ausgewählte Optimierungsprobleme und Algorithmen auf gegnerisch vorgegebenen Instanzen, die durch ein leichtes Zufallsrauschen gestört werden. Solche perturbierten Instanzen bezeichnen wir als semi-zufällige Eingaben. Als Beispiel betrachten wir den Successive- Shortest-Path-Algorithmus für das Minimum-Cost-Flow-Problem. Während dieser Algorithmus imWorst Case exponentiell viele Schritte benötigt, um einen Minimum-Cost-Flow zu berechnen, zeigen wir, dass seine Laufzeit auf semi-zufälligen Eingaben polynomiell ist. Ein weiteres Problem, das wir in dieser Arbeit untersuchen, ist die Minimierung des Makespans für Scheduling auf unterschiedlich schnellen Maschinen. Es scheint, dass dieses Problem nicht effizient gelöst werden kann. Daher betrachten wir drei Approximationsalgorithmen: den Jump-, den Lex-Jump- und den List-Scheduling-Algorithmus. Im Worst Case hängt die Approximationsgüte dieser Algorithmen von der Anzahl der Maschinen ab. Wir zeigen, dass das auf semi-zufälligen Eingaben nicht der Fall ist. Des Weiteren betrachten wir ganzzahlige Optimierungsprobleme mit mehreren linearen Zielfunktionen, die simultan minimiert werden sollen. Wir leiten eine polynomielle obere Schranke für die Größe der Pareto-Menge auf semi-zufälligen Eingaben her, die im Gegensatz zu der exponentiellen unteren Worst-Case-Schranke steht. Mit den Erkenntnissen aus der Laufzeitanalyse des Successive-Shortest-Path-Algorithmus entwerfen wir einen randomisierten Algorithmus zur Bestimmung eines kurzen Pfades zwischen zwei gegebenen Ecken eines Polyeders. Wir betrachten dieses Ergebnis als ein Indiz dafür, dass in Zukunft Analysen auf semi-zufälligen Eingaben auch zu der Entwicklung schneller Algorithmen führen könnten

Smoothed Analysis of the Successive Shortest Path Algorithm

The minimum-cost flow problem is a classic problem in combinatorial optimization with various applications. Several pseudo-polynomial, polynomial, and strongly polynomial algorithms have been developed in the past decades, and it seems that both the problem and the algorithms are well understood. However, some of the algorithms' running times observed in empirical studies contrast the running times obtained by worst-case analysis not only in the order of magnitude but also in the ranking when compared to each other. For example, the Successive Shortest Path (SSP) algorithm, which has an exponential worst-case running time, seems to outperform the strongly polynomial Minimum-Mean Cycle Canceling algorithm. To explain this discrepancy, we study the SSP algorithm in the framework of smoothed analysis and establish a bound of $O(mn\phi)$ for the number of iterations, which implies a smoothed running time of $O(mn\phi (m + n\log n))$, where $n$ and $m$ denote the number of nodes and edges, respectively, and $\phi$ is a measure for the amount of random noise. This shows that worst-case instances for the SSP algorithm are not robust and unlikely to be encountered in practice. Furthermore, we prove a smoothed lower bound of $\Omega(m \phi \min\{n, \phi\})$ for the number of iterations of the SSP algorithm, showing that the upper bound cannot be improved for $\phi = \Omega(n)$.Comment: A preliminary version has been presented at SODA 201

Learning new sensorimotor contingencies:Effects of long-term use of sensory augmentation on the brain and conscious perception

Theories of embodied cognition propose that perception is shaped by sensory stimuli and by the actions of the organism. Following sensorimotor contingency theory, the mastery of lawful relations between own behavior and resulting changes in sensory signals, called sensorimotor contingencies, is constitutive of conscious perception. Sensorimotor contingency theory predicts that, after training, knowledge relating to new sensorimotor contingencies develops, leading to changes in the activation of sensorimotor systems, and concomitant changes in perception. In the present study, we spell out this hypothesis in detail and investigate whether it is possible to learn new sensorimotor contingencies by sensory augmentation. Specifically, we designed an fMRI compatible sensory augmentation device, the feelSpace belt, which gives orientation information about the direction of magnetic north via vibrotactile stimulation on the waist of participants. In a longitudinal study, participants trained with this belt for seven weeks in natural environment. Our EEG results indicate that training with the belt leads to changes in sleep architecture early in the training phase, compatible with the consolidation of procedural learning as well as increased sensorimotor processing and motor programming. The fMRI results suggest that training entails activity in sensory as well as higher motor centers and brain areas known to be involved in navigation. These neural changes are accompanied with changes in how space and the belt signal are perceived, as well as with increased trust in navigational ability. Thus, our data on physiological processes and subjective experiences are compatible with the hypothesis that new sensorimotor contingencies can be acquired using sensory augmentation