Counting Markov Equivalent Directed Acyclic Graphs Consistent with Background Knowledge

TL;DR

This paper extends existing algorithms to count DAGs in Markov equivalence classes with fixed edge directions, demonstrating fixed-parameter tractability in certain cases, especially with partial background knowledge.

Contribution

It introduces a fixed-parameter tractable algorithm for counting DAGs with fixed edges, generalizing previous polynomial-time methods.

Findings

Counting remains tractable with fixed background knowledge in certain cases.

The algorithm's runtime is polynomial with degree independent of additional fixed edges.

The problem is generally hard but becomes manageable under specific parameters.

Abstract

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 \emph{not} depend upon the number of additional edges provided as input.