BDDs in a branch and cut framework

Branch and cut is today's state-of-the-art method to solve 0/1-integer linear programs. Important for the success of this method is the generation of strong valid inequalities, which tighten the linear programming relaxation of 0/1-IPs and thus allow for early pruning of parts of the search tree. In this paper we present a novel approach to generate valid inequalities for 0/1-IPs which is based on binary decision diagrams (BDDs). BDDs are a data-structure which represents 0/1-vectors as paths of a certain acyclic graph. They have been successfully applied in computational logic, hardware verification and synthesis. We implemented our BDD cutting plane generator in a branch-and-cut framework and tested it on several instances of the MAX-ONES problem and randomly generated 0/1-IPs. Our computational results show that we have developed competitive code for these problems, on which state-of-the-art MIP-solvers fall shor

Binary Decision Diagrams and Integer Programming

In this work we show how Binary Decision Diagrams can be used as a powerful tool for 0/1 Integer Programming and related polyhedral problems. We develop an output-sensitive algorithm for building a threshold BDD, which represents the feasible 0/1 solutions of a linear constraint, and give a parallel and-operation for threshold BDDs to build the BDD for a 0/1 IP. In addition we construct a 0/1 IP for nding the optimal variable order and computing the variable ordering spectrum of a threshold BDD. For the investigation of the polyhedral structure of a 0/1 IP we show how BDDs can be applied to count or enumerate all 0/1 vertices of the corresponding 0/1 polytope, enumerate its facets, and nd an optimal solution or count or enumerate all optimal solutions to a linear objective function. Furthermore we developed the freely available tool azove which outperforms existing codes for the enumeration of 0/1 points. Branch & Cut is today's state-of-the-art method to solve 0/1 IPs. We present a novel approach to generate valid inequalities for 0/1 IPs which is based on BDDs. We implemented our BDD based separation routine in a B&C framework. Our computationa

5 statistische Aufsätze über Effizienz und Produktivität. Eine Anwendung der Data Envelopment Analysis in Wirtschaftsgeschichte

This thesis uses the Data Envelopment Analysis to analyze topics in economic history. All parts are concerned with productivity and efficiency. For the United States in the nineteenth century, it is shown that factories were more efficient than artisanal shops, and the hypothesis that regional efficiency differences were responsible for different industrial development paths cannot be confirmed. It is also shown that the american civil war had rather less influence on manufacturing efficiency. The war-torn regions were somewhat stronger after the war in technical efficiency, while they were worse in scale efficiency. Analyzing the determinants of labor productivity in Russia the results suggest a strong positive influence of nutrition and education in the 1870s on labor productivity. Political influences proved to be important. When looking at a global dataset of up to 62 countries from the 1850s to the 1980s estimates of government efficiency are presented and the correlates analyzed. Results suggest that some highly developed countries were remarkably deficiency with regard to efficiency and that ethnic heterogeneity lowers efficiency. Altogether, this thesis shows the potential of the data envelopment analysis to shed light on controversial topics in economic history.Die Arbeit nutzt die Date Envelopment Analysis, ein bisher wenig genutzte Methode in der Wirtschaftsgeschichte, um Produktivitä und Effizienz genauer zu untersuchen. Es zeigt sich, dass diese Methode zu einigen kontroversen Diskussionen der Wirtschaftsgeschichte neue Erkenntnisse beitragen kann. Exemplarisch sei die Diskussion um die Gründe für die unterschiedliche regionale industrielle Entwicklung in den Vereinigten Staaten im 19. Jahrhundert genannt. Die vorliegenden Ergebnisse deuten darauf hin dass es nur geringe regionale Effizienzvariation gab und diese die wirtschaftliche Entwicklung nicht erklären können. Außerdem zeigen sich Fabriken als effizienter als kleine Handwerksbetriebe. Mit einem großen Datensatz mit bis zu 62 Ländern zwischen 1850 und 1980 werden neue Trends für staatliche Effizienzentwicklungen präsentiert und analysiert. Dabei zeigt sich dass politische Instabilität und eine ethnisch heterogene Bevölkerung die staatliche Effizienz senken

0/1 vertex and facet enumeration with BDDs

In polyhedral studies of 0/1 polytopes two prominent problems exist. One is the vertex enumeration problem: Given a system of inequalities, enumerate its feasible 0/1 points. Another one is the convex hull problem: Given a set of 0/1 points in dimension d, enumerate the facets of the corresponding polytope. We present two new approaches for both problems. The novelty of our algorithms is the incorporation of binary decision diagrams (BDDs), a datastructure which has become very popular and effective in hardware verification and computational logic. Our computational results show the strength of our methods. We introduce our new tool azove which is currently the fastest software for counting and enumerating 0/1 points in a polytope

A Primal Branch-and-Cut Algorithm for the Degree-Constrained Minimum Spanning Tree Problem

The degree-constrained minimum spanning tree (DCMST) is relevant in the design of networks. It consists of finding a spanning tree whose nodes do not exceed a given maximum degree and whose total edge length is minimum. We design a primal branch-and-cut algorithm that solves instances of the problem to optimality. Primal methods have not been used extensively in the past, and their performance often could not compete with their standard ‘dual’ counterparts. We show that primal separation procedures yield good bounds for the DCMST problem. On several instances, the primal branch-and-cut program turns out to be competitive with other methods known in the literature. This shows the potential of the primal method

BDDs in a branch & cut framework

Branch & Cut is today’s state-of-the-art method to solve 0/1-integer linear programs. Important for the success of this method is the generation of strong valid inequalities, which tighten the linear programming relaxation of 0/1-IPs and thus allow for early pruning of parts of the search tree. In this paper we present a novel approach to generate valid inequalities for 0/1-IPs which is based on Binary Decision Diagrams (BDDs). BDDs are a datastructure which represents 0/1-vectors as paths of a certain acyclic graph. They have been successfully applied in computational logic, hardware verification and synthesis. We implemented our BDD cutting plane generator in a branch-and-cut framework and tested it on several instances of the MAX-ONES problem and randomly generated 0/1-IPs. Our computational results show that we have developed competitive code for these problems, on which state-of-the-art MIP-solvers fall short