Generation of random chordal graphs using subtrees of a tree

Chordal graphs form one of the most studied graph classes. Several graph problems that are NP-hard in general become solvable in polynomial time on chordal graphs, whereas many others remain NP-hard. For a large group of problems among the latter, approximation algorithms, parameterized algorithms, and algorithms with moderately exponential or sub-exponential running time have been designed. Chordal graphs have also gained increasing interest during the recent years in the area of enumeration algorithms. Being able to test these algorithms on instances of chordal graphs is crucial for understanding the concepts of tractability of hard problems on graph classes. Unfortunately, only few published papers give algorithms for generating chordal graphs. Even in these papers, only very few methods aim for generating a large variety of chordal graphs. Surprisingly, none of these methods is directly based on the “intersection of subtrees of a tree” characterization of chordal graphs. In this paper, we give an algorithm for generating chordal graphs, based on the characterization that a graph is chordal if and only if it is the intersection graph of subtrees of a tree. Upon generating a random host tree, we give and test various methods that generate subtrees of the host tree. We compare our methods to one another and to existing ones for generating chordal graphs. Our experiments show that one of our methods is able to generate the largest variety of chordal graphs in terms of maximal clique sizes. Moreover, two of our subtree generation methods result in an overall complexity of our generation algorithm that is the best possible time complexity for a method generating the entire node set of subtrees in an “intersection of subtrees of a tree” representation. The instances corresponding to the results presented in this paper, and also a set of relatively small-sized instances are made available online.publishedVersio

Clique trees of infinite locally finite chordal graphs

We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting

Sequential sampling of junction trees for decomposable graphs

The junction-tree representation provides an attractive structural property for organizing a decomposable graph. In this study, we present a novel stochastic algorithm, which we call the junction-tree expander, for sequential sampling of junction trees for decomposable graphs. We show that recursive application of the junction-tree expander, expanding incrementally the underlying graph with one vertex at a time, has full support on the space of junction trees with any given number of underlying vertices. A direct application of our suggested algorithm is demonstrated in a sequential Monte Carlo setting designed for sampling from distributions on spaces of decomposable graphs, where the junction-tree expander can be effectively employed as proposal kernel; see the companion paper Olsson et al. 2019 [16]. A numerical study illustrates the utility of our approach by two examples: in the first one, how the junction-tree expander can be incorporated successfully into a particle Gibbs sampler for Bayesian structure learning in decomposable graphical models; in the second one, we provide an unbiased estimator of the number of decomposable graphs for a given number of vertices. All the methods proposed in the paper are implemented in the Python library trilearn.Comment: 31 pages, 7 figure

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

Constant tolerance intersection graphs of subtrees of a tree

AbstractA chordal graph is the intersection graph of a family of subtrees of a host tree. In this paper we generalize this. A graph G=(V,E) has an (h,s,t)-representation if there exists a host tree T of maximum degree at most h, and a family of subtrees {Sv}v∈V of T, all of maximum degree at most s, such that uv∈E if and only if |Su∩Sv|⩾t. For given h,s, and t, there exist infinitely many forbidden induced subgraphs for the class of (h,s,t)-graphs. On the other hand, for fixed h⩾s⩾3, every graph is an (h,s,t)-graph provided that we take t large enough. Under certain conditions representations of larger graphs can be obtained from those of smaller ones by amalgamation procedures. Other representability and non-representability results are presented as well