Playing Billiard in Version Space

A ray-tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a set of linear separable examples. While the Bayes-optimum requires a majority decision over all Perceptrons separating the example set, the problem considered here corresponds to finding the single Perceptron with best average generalization probability. For randomly distributed examples the billiard estimate agrees with known analytic results. In real-life classification problems the generalization error is consistently reduced compared to the maximal stability Perceptron.Comment: uuencoded, gzipped PostScript file, 127576 bytes To recover 1) save file as bayes.uue. Then 2) uudecode bayes.uue and 3) gunzip bayes.ps.g

Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming

Here we study the NP-complete $K$-SAT problem. Although the worst-case complexity of NP-complete problems is conjectured to be exponential, there exist parametrized random ensembles of problems where solutions can typically be found in polynomial time for suitable ranges of the parameter. In fact, random $K$-SAT, with $\alpha=M/N$ as control parameter, can be solved quickly for small enough values of $\alpha$. It shows a phase transition between a satisfiable phase and an unsatisfiable phase. For branch and bound algorithms, which operate in the space of feasible Boolean configurations, the empirically hardest problems are located only close to this phase transition. Here we study $K$-SAT ($K=3,4$) and the related optimization problem MAX-SAT by a linear programming approach, which is widely used for practical problems and allows for polynomial run time. In contrast to branch and bound it operates outside the space of feasible configurations. On the other hand, finding a solution within polynomial time is not guaranteed. We investigated several variants like including artificial objective functions, so called cutting-plane approaches, and a mapping to the NP-complete vertex-cover problem. We observed several easy-hard transitions, from where the problems are typically solvable (in polynomial time) using the given algorithms, respectively, to where they are not solvable in polynomial time. For the related vertex-cover problem on random graphs these easy-hard transitions can be identified with structural properties of the graphs, like percolation transitions. For the present random $K$-SAT problem we have investigated numerous structural properties also exhibiting clear transitions, but they appear not be correlated to the here observed easy-hard transitions. This renders the behaviour of random $K$-SAT more complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure

Unsupervised Video Understanding by Reconciliation of Posture Similarities

Understanding human activity and being able to explain it in detail surpasses mere action classification by far in both complexity and value. The challenge is thus to describe an activity on the basis of its most fundamental constituents, the individual postures and their distinctive transitions. Supervised learning of such a fine-grained representation based on elementary poses is very tedious and does not scale. Therefore, we propose a completely unsupervised deep learning procedure based solely on video sequences, which starts from scratch without requiring pre-trained networks, predefined body models, or keypoints. A combinatorial sequence matching algorithm proposes relations between frames from subsets of the training data, while a CNN is reconciling the transitivity conflicts of the different subsets to learn a single concerted pose embedding despite changes in appearance across sequences. Without any manual annotation, the model learns a structured representation of postures and their temporal development. The model not only enables retrieval of similar postures but also temporal super-resolution. Additionally, based on a recurrent formulation, next frames can be synthesized.Comment: Accepted by ICCV 201

Practical Minimum Cut Algorithms

The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our algorithm is based on cluster contraction using label propagation and Padberg and Rinaldi's contraction heuristics [SIAM Review, 1991]. We give both sequential and shared-memory parallel implementations of our algorithm. Extensive experiments on both real-world and generated instances show that our algorithm finds the optimal cut on nearly all instances significantly faster than other state-of-the-art algorithms while our error rate is lower than that of other heuristic algorithms. In addition, our parallel algorithm shows good scalability

Using the TSP Solution for Optimal Route Scheduling in Construction Management

This paper presents the optimal route scheduling in construction management by using the solution of the traveling salesman problem (TSP). The TSP is a well-known combinatorial optimization problem which holds a considerable potential for applications in construction management. The aim of this paper is to bring forward the solution of the TSP to the wider expert community. For this purpose, the TSP model formulation, the applicability of the TSP optimization model and the commercially available software for modelling and solving the TSP are presented. An example of the optimal route scheduling by using the solution of the TSP is demonstrated at the end of the paper to show the applicability of the TSP model

Diseño de un modelo para un problema de distribución de tuberías, con entregas divididas y flota heterogénea.

OBJETIVO Y MÉTODO DE ESTUDIO En esta tesis se describe el problema de ruteo de vehículos de una empresa regiomontana de tubería ligera. La empresa debe distribuir desde cualesquiera de sus 5 plantas sus productos a un grupo determinado de clientes que se encuentran dispersos en toda la República Mexicana, actividad que se realiza en base a la experiencia del encargado de ruteo y no en un sistema de optimización el cual evitaría el uso ineficiente de recursos y generaría grandes ahorros en el costo de transportación y por lo tanto en el costo logístico de la empresa. El Problema de Ruteo de Vehículos (VRP) se refiere al grupo de problemas que abordan la distribución a un conjunto de clientes dispersos geográficamente utilizando una flota de vehículos. Su finalidad es encontrar un camino, o ruta, que recorra todos los clientes minimizando los costos relacionados con el recorrido satisfaciendo cierto número de restricciones. El VRP es un problema de optimización que se puede encontrar en diversas situaciones de la vida real, ya sea en la industria, en los servicios o en el mismo vivir de las personas. Uno de los propósitos de esta tesis es presentar un modelo de optimización basado en las características particulares de la empresa y de su sistema de transporte como lo son: uso de flota heterogénea, múltiples productos, entregas divididas. Otro de los objetivos es probar con la ayuda del software de optimización los ahorros que se generarían al integrar dicho modelo como parte de sus operaciones en el ruteo de vehículos