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1. Counting and Sampling Orientations on Chordal Graphs

   https://doi.org/10.1007/978-3-030-96731-4_29  

   Bezakova, Ivona; Sun, Wenbo (January 2022, International Conference and Workshops on Algorithms and Computation (WALCOM))

   We study counting problems for several types of orientations of chordal graphs: source-sink-free orientations, sink-free orientations, acyclic orientations, and bipolar orientations, and, for the latter two, we also present linear-time uniform samplers. Counting sink-free, acyclic, or bipolar orientations are known to be #P-complete for general graphs, motivating our study on a restricted, yet well-studied, graph class. Our main focus is source-sink-free orientations, a natural restricted version of sink-free orientations related to strong orientations, which we introduce in this work. These orientations are intriguing, since despite their similarity, currently known FPRAS and sampling techniques (such as Markov chains or sink-popping) that apply to sink-free orientations do not seem to apply to source-sink-free orientations. We present fast polynomialtime algorithms counting these orientations on chordal graphs. Our approach combines dynamic programming with inclusion-exclusion (going two levels deep for source-sink-free orientations and one level for sinkfree orientations) throughout the computation. Dynamic programming counting algorithms can be typically used to produce a uniformly random sample. However, due to the negative terms of the inclusion-exclusion, the typical approach to obtain a polynomial-time sampling algorithm does not apply in our case. Obtaining such an almost uniform sampling algorithm for source-sink-free orientations in chordal graphs remains an open problem. Little is known about counting or sampling of acyclic or bipolar orientations, even on restricted graph classes. We design efficient (linear-time) exact uniform sampling algorithms for these orientations on chordal graphs. These algorithms are a byproduct of our counting algorithms, but unlike in other works that provide dynamic-programming-based samplers, we produce a random orientation without computing the corresponding count, which leads to a faster running time than the counting algorithm (since it avoids manipulation of large integers). 

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2. Sampling Partial Acyclic Orientations in Chordal Graphs by the Lovasz Local Lemma (Student Abstract)

   Sun, W; Bezakova, I (January 2021, Thirty-Fifth AAAI Conference on Artificial Intelligence)

   null (Ed.)

   Sampling of various types of acyclic orientations of chordal graphs plays a central role in several AI applications. In this work we investigate the use of the recently proposed general partial rejection sampling technique of Guo, Jerrum, and Liu, based on the Lovasz Local Lemma, for sampling partial acyclic orientations. For a given undirected graph, an acyclic orientation is an assignment of directions to all of its edges so that there is no directed cycle. In partial orientations some edges are allowed to be undirected. We show how the technique can be used to sample partial acyclic orientations of chordal graphs fast and with a clearly specified underlying distribution. This is in contrast to other samplers of various acyclic orientations with running times exponentially dependent on the maximum degree of the graph. 

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3. Counting and Sampling Labeled Chordal Graphs in Polynomial Time

   https://doi.org/10.4230/LIPIcs.ESA.2023.58  

   Hébert-Johnson, Úrsula; Lokshtanov, Daniel; Vigoda, Eric (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J; Herman, Grzegorz (Ed.)

   We present the first polynomial-time algorithm to exactly compute the number of labeled chordal graphs on n vertices. Our algorithm solves a more general problem: given n and ω as input, it computes the number of ω-colorable labeled chordal graphs on n vertices, using O(n⁷) arithmetic operations. A standard sampling-to-counting reduction then yields a polynomial-time exact sampler that generates an ω-colorable labeled chordal graph on n vertices uniformly at random. Our counting algorithm improves upon the previous best result by Wormald (1985), which computes the number of labeled chordal graphs on n vertices in time exponential in n. An implementation of the polynomial-time counting algorithm gives the number of labeled chordal graphs on up to 30 vertices in less than three minutes on a standard desktop computer. Previously, the number of labeled chordal graphs was only known for graphs on up to 15 vertices. 

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4. Sampling Random Chordal Graphs by MCMC (Student Abstract)

   https://doi.org/10.1609/aaai.v34i10.7237  

   Sun, Wenbo; Bezáková, Ivona (June 2020, Proceedings of the AAAI Conference on Artificial Intelligence)

   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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5. A Dichotomy Theorem for Linear Time Homomorphism Orbit Counting in Bounded Degeneracy Graphs

   https://doi.org/10.4230/LIPICS.ISAAC.2024.54  

   Paul-Pena, Daniel; Seshadhri, C (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mestre, Julián; Wirth, Anthony (Ed.)

   Counting the number of homomorphisms of a pattern graph H in a large input graph G is a fundamental problem in computer science. In many applications in databases, bioinformatics, and network science, we need more than just the total count. We wish to compute, for each vertex v of G, the number of H-homomorphisms that v participates in. This problem is referred to as homomorphism orbit counting, as it relates to the orbits of vertices of H under its automorphisms. Given the need for fast algorithms for this problem, we study when near-linear time algorithms are possible. A natural restriction is to assume that the input graph G has bounded degeneracy, a commonly observed property in modern massive networks. Can we characterize the patterns H for which homomorphism orbit counting can be done in near-linear time? We discover a dichotomy theorem that resolves this problem. For pattern H, let 𝓁 be the length of the longest induced path between any two vertices of the same orbit (under the automorphisms of H). If 𝓁 ≤ 5, then H-homomorphism orbit counting can be done in near-linear time for bounded degeneracy graphs. If 𝓁 > 5, then (assuming fine-grained complexity conjectures) there is no near-linear time algorithm for this problem. We build on existing work on dichotomy theorems for counting the total H-homomorphism count. Surprisingly, there exist (and we characterize) patterns H for which the total homomorphism count can be computed in near-linear time, but the corresponding orbit counting problem cannot be done in near-linear time. 

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