# Counting Markov Equivalent Directed Acyclic Graphs Consistent with Background Knowledge

arXiv: https://arxiv.org/abs/2206.06744 · HTML: https://mlsanity.com/paper/2206.06744/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 30 references that are in the map.
- First posted on arXiv 14 Jun 2022.

## Builds on (most cited first, 6 of 6)

- [Causal Inference and Causal Explanation with Background Knowledge](https://mlsanity.com/paper/1302.4972/) (2013)
- [A Transformational Characterization of Equivalent Bayesian Network Structures](https://mlsanity.com/paper/1302.4938/) (2013)
- [Graphical Models for Game Theory](https://mlsanity.com/paper/1301.2281/) (2013)
- [Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs with Applications](https://mlsanity.com/paper/2205.02654/) (2022)
- [Counting and Sampling from Markov Equivalent DAGs Using Clique Trees](https://mlsanity.com/paper/1802.01239/) (2018)
- [Formulas for Counting the Sizes of Markov Equivalence Classes of Directed Acyclic Graphs](https://mlsanity.com/paper/1610.07921/) (2016)

## Abstract (arXiv)

A polynomial-time exact algorithm for counting the number of directed acyclic graphs in a Markov equivalence class was recently given by Wien"obst, Bannach, and Li'skiewicz (AAAI 2021). In this paper, we consider the more general problem of counting the number of directed acyclic graphs in a Markov equivalence class when the directions of some of the edges are also fixed (this setting arises, for example, when interventional data is partially available). This problem has been shown in earlier work to be complexity-theoretically hard. In contrast, we show that the problem is nevertheless tractable in an interesting class of instances, by establishing that it is ``fixed-parameter tractable''. In particular, our counting algorithm runs in time that is bounded by a polynomial in the size of the graph, where the degree of the polynomial does not depend upon the number of additional edges provided as input.