Efficient Algorithms for Probabilistic Inference, Combinatorial Optimization and the Discovery of Causal Structure from Data
In the first article we present a network based algorithm for probabilistic inference in an undirected structure. We show that the algorithm can be used as a general purpose approximation algorithm for combinatorial optimization, and discuss issues of approximation and convergence. We show the successful use of the algorithm for the assignment problem, the queens problem, and the set covering problem.
In the second article, we develop a version of this algorithm for 0-1 integer programming problems when A is 0-1 and b is integer. This version can be described in terms of an ascent algorithm for the LP-relaxation, and an approximation scheme working together with this method. The algorithm features a parameter that determines the degree of approximation. We show that the algorithm performs well for set covering problems compared to the CPLEX system.
In the third article, we present an algorithm to determine the interaction structure in a multidimensional binary sample. We show that the algorithm is capable of reconstructing most of the causal structure for problems with more than a hundred variables, including many of the causal directions. The algorithm is based on an efficient method for approximate ML-estimation in a given undirected structure. When the structure is not known, the MDL-criterion is used to determine the goodness of a hypothesis, and the structure is determined incrementally by a search algorithm, working together with the parameter estimation algorithm.