Probabilistic Methodology and Techniques for Artefact Conception and Development

The purpose of this paper is to make a state of the art on probabilistic methodology and techniques for artefact conception and development. It is the 8th deliverable of the BIBA (Bayesian Inspired Brain and Artefacts) project. We first present the incompletness problem as the central difficulty that both living creatures and artefacts have to face: how can they perceive, infer, decide and act efficiently with incomplete and uncertain knowledge?. We then introduce a generic probabilistic formalism called Bayesian Programming. This formalism is then used to review the main probabilistic methodology and techniques. This review is organized in 3 parts: first the probabilistic models from Bayesian networks to Kalman filters and from sensor fusion to CAD systems, second the inference techniques and finally the learning and model acquisition and comparison methodologies. We conclude with the perspectives of the BIBA project as they rise from this state of the art

Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time

It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in terms of the weights). We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms, also for weighted and counting versions. For example, in this time we can solve the traveling salesman problem or count the number of Hamiltonian cycles. The rank-based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying "small" sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the matrix tree theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page

Answer Set Solving with Bounded Treewidth Revisited

Parameterized algorithms are a way to solve hard problems more efficiently, given that a specific parameter of the input is small. In this paper, we apply this idea to the field of answer set programming (ASP). To this end, we propose two kinds of graph representations of programs to exploit their treewidth as a parameter. Treewidth roughly measures to which extent the internal structure of a program resembles a tree. Our main contribution is the design of parameterized dynamic programming algorithms, which run in linear time if the treewidth and weights of the given program are bounded. Compared to previous work, our algorithms handle the full syntax of ASP. Finally, we report on an empirical evaluation that shows good runtime behaviour for benchmark instances of low treewidth, especially for counting answer sets.Comment: This paper extends and updates a paper that has been presented on the workshop TAASP'16 (arXiv:1612.07601). We provide a higher detail level, full proofs and more example