Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs

Wienöbst, Marcel; Bannach, Max; Liśkiewicz, Maciej

doi:10.48550/arxiv.2012.09679

Cited by 2 publications

(3 citation statements)

References 22 publications

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“…Towards the proof of this theorem, we note rst that the existence of a topological ordering 𝜎 satisfying just P1 can be established using ideas from the analysis of, e.g., the "maximum cardinality search" algorithm for chordal graphs (Tarjan and Yannakakis (1984), see also Corollary 2 of Wienöbst et al (2021)). We state this here as a lemma, and provide the proof in Section A.…”

Section: Preliminariesmentioning

confidence: 99%

“…Proof. As already alluded to in the main paper, the proof of item 1 uses ideas that are very similar to the "maximal cardinality search" algorithm for chordal graphs (Tarjan and Yannakakis (1984), see also Corollary 2 of Wienöbst et al (2021)). Fix an arbitrary topological ordering 𝜏 of 𝐷, and let 𝑣 1 = 𝜏 (1) be the top vertex in 𝜏.…”

Section: A Proof Of Lemma 34 Of the Main Papermentioning

confidence: 99%

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Universal Lower Bound for Learning Causal DAGs with Atomic Interventions

Porwal¹,

Srivastava²,

Sinha³

2021

Preprint

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A well-studied challenge that arises in the structure learning problem of causal directed acyclic graphs (DAG) is that using observational data, one can only learn the graph up to a "Markov equivalence class" (MEC). The remaining undirected edges have to be oriented using interventions, which can be very expensive to perform in applications. Thus, the problem of minimizing the number of interventions needed to fully orient the MEC has received a lot of recent attention, and is also the focus of this work. We prove two main results. The rst is a new universal lower bound on the number of atomic interventions that any algorithm (whether active or passive) would need to perform in order to orient a given MEC. Our second result shows that this bound is, in fact, within a factor of two of the size of the smallest set of atomic interventions that can orient the MEC. Our lower bound is provably better than previously known lower bounds. The proof of our lower bound is based on the new notion of clique-block shared-parents (CBSP) orderings, which are topological orderings of DAGs without v-structures and satisfy certain special properties. Further, using simulations on synthetic graphs and by giving examples of special graph families, we show that our bound is often signi cantly better.

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Section: Preliminariesmentioning

confidence: 99%

Section: A Proof Of Lemma 34 Of the Main Papermentioning

confidence: 99%

Universal Lower Bound for Learning Causal DAGs with Atomic Interventions

Porwal¹,

Srivastava²,

Sinha³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Sampling and counting of different types of acyclic orientations over chordal graphs attracted attention in several AI research areas, for example in structure learning of Bayesian networks (Ganian, Hamm, and Talvitie 2020;Ghassami et al 2019;Talvitie and Koivisto 2019;Wienöbst, Bannach, and Liskiewicz 2020). A graph is chordal if each of its cycles of length at least four has an edge that connects two nonadjacent vertices in that cycle.…”

Section: Introductionmentioning

confidence: 99%

Sampling and Counting Acyclic Orientations in Chordal Graphs (Student Abstract)

Sun

2022

AAAI

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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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Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs

Cited by 2 publications

References 22 publications

Universal Lower Bound for Learning Causal DAGs with Atomic Interventions

Universal Lower Bound for Learning Causal DAGs with Atomic Interventions

Sampling and Counting Acyclic Orientations in Chordal Graphs (Student Abstract)

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