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Sampling of chordal graphs and various types of acyclic orientations over chordal graphs plays a central role in several AI applications such as causal structure learning. For a given undirected graph, an acyclic orientation is an assignment of directions to all of its edges which makes the resulting directed graph cycle-free. Sampling is often closely related to the corresponding counting problem. Counting of acyclic orientations of a given chordal graph can be done in polynomial time, but the previously known techniques do not seem to lead to a corresponding (efficient) sampler. In this work, we propose a dynamic programming framework which yields a counter and a uniform sampler, both of which run in (essentially) linear time. An interesting feature of our sampler is that it is a stand-alone algorithm that, unlike other DP-based samplers, does not need any preprocessing which determines the corresponding counts.
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Counting and Sampling Orientations on Chordal Graphs
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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- PAR ID:
- 10355288
- Date Published:
- Journal Name:
- International Conference and Workshops on Algorithms and Computation (WALCOM)
- Volume:
- 13174
- Page Range / eLocation ID:
- 352-364
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-96731-4_29
