Approximate Counting, the Lovasz Local Lemma and Inference in Graphical Models

In this paper we introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula $\Phi$ when the width is logarithmic in the maximum degree. This closes an exponential gap between the known upper and lower bounds. Moreover our algorithm extends straightforwardly to approximate sampling, which shows that under Lov\'asz Local Lemma-like conditions it is not only possible to find a satisfying assignment, it is also possible to generate one approximately uniformly at random from the set of all satisfying assignments. Our approach is a significant departure from earlier techniques in approximate counting, and is based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables. Finally, we give an application of our results to show that it is algorithmically possible to sample from the posterior distribution in an interesting class of graphical models.Comment: 25 pages, 2 figure

Perfect Sampling for Hard Spheres from Strong Spatial Mixing

We provide a perfect sampling algorithm for the hard-sphere model on subsets of R^d with expected running time linear in the volume under the assumption of strong spatial mixing. A large number of perfect and approximate sampling algorithms have been devised to sample from the hard-sphere model, and our perfect sampling algorithm is efficient for a range of parameters for which only efficient approximate samplers were previously known and is faster than these known approximate approaches. Our methods also extend to the more general setting of Gibbs point processes interacting via finite-range, repulsive potentials

Top-k Querying of Unknown Values under Order Constraints

Many practical scenarios make it necessary to evaluate top-k queries over data items with partially unknown values. This paper considers a setting where the values are taken from a numerical domain, and where some partial order constraints are given over known and unknown values: under these constraints, we assume that all possible worlds are equally likely. Our work is the first to propose a principled scheme to derive the value distributions and expected values of unknown items in this setting, with the goal of computing estimated top-k results by interpolating the unknown values from the known ones. We study the complexity of this general task, and show tight complexity bounds, proving that the problem is intractable, but can be tractably approximated. We then consider the case of tree-shaped partial orders, where we show a constructive PTIME solution. We also compare our problem setting to other top-k definitions on uncertain data

Efficient Sampling and Counting of Graph Structures related to Chordal Graphs

Counting problems aim to count the number of solutions for a given input, for example, counting the number of variable assignments that satisfy a Boolean formula. Sampling problems aim to produce a random object from a desired distribution, for example, producing a variable assignment drawn uniformly at random from all assignments that satisfy a Boolean formula. The problems of counting and sampling of graph structures on different types of graphs have been studied for decades for their great importance in areas like complexity theory and statistical physics. For many graph structures such as independent sets and acyclic orientations, it is widely believed that no exact or approximate (with an arbitrarily small error) polynomial-time algorithms on general graphs exist. Therefore, the research community studies various types of graphs, aiming either to design a polynomial-time counting or sampling algorithm for such inputs, or to prove a corresponding inapproximability result. Chordal graphs have been studied widely in both AI and theoretical computer science, but their study from the counting perspective has been relatively limited. Previous works showed that some graph structures can be counted in polynomial time on chordal graphs, when their counting on general graphs is provably computationally hard. The main objective of this thesis is to design and analyze counting and sampling algorithms for several well-known graph structures, including independent sets and different types of graph orientations, on chordal graphs. Our contributions can be described from two perspectives: evaluating the performances of some well-known sampling techniques, such as Markov chain Monte Carlo, on chordal graphs; and showing that the chordality does make those counting problems polynomial-time solvable