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Chordal graphs are a widely studied graph class, with applications in several areas of computer science, including structural learning of Bayesian networks. Many problems that are hard on general graphs become solvable on chordal graphs. The random generation of instances of chordal graphs for testing these algorithms is often required. Nevertheless, there are only few known algorithms that generate random chordal graphs, and, as far as we know, none of them generate chordal graphs uniformly at random (where each chordal graph appears with equal probability). In this paper we propose a Markov chain Monte Carlo (MCMC) method to sample connected chordal graphs uniformly at random. Additionally, we propose a Markov chain that generates connected chordal graphs with a bounded treewidth uniformly at random. Bounding the treewidth parameter (which bounds the largest clique) has direct implications on the running time of various algorithms on chordal graphs. For each of the proposed Markov chains we prove that they are ergodic and therefore converge to the uniform distribution. Finally, as initial evidence that the Markov chains have the potential to mix rapidly, we prove that the chain on graphs with bounded treewidth mixes rapidly for trees (chordal graphs with treewidth bound of one).
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Projections of the Aldous chain on binary trees: Intertwining and consistency
Consider the Aldous Markov chain on the space of rooted binary trees withnlabeled leaves in which at each transition a uniform random leaf is deleted and reattached to a uniform random edge. Now, fix 1 ≤ k<nand project the leaf mass onto the subtree spanned by the firstkleaves. This yields a binary tree with edge weights that we call a “decoratedk‐tree with total massn.” We introduce label swapping dynamics for the Aldous chain so that, when it runs in stationarity, the decoratedk‐trees evolve as Markov chains themselves, and are projectively consistent overk. The construction of projectively consistent chains is a crucial step in the construction of the Aldous diffusion on continuum trees by the present authors, which is then→∞continuum analog of the Aldous chain and will be taken up elsewhere.
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- Award ID(s):
- 1855568
- PAR ID:
- 10155845
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 57
- Issue:
- 3
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 745-769
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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