Faster Fixed Parameter Tractable Algorithms for Counting Markov Equivalence Classes with Special Skeletons
Vidya Sagar Sharma
TL;DR
The paper addresses counting Markov Equivalence Classes with a fixed skeleton graph $G$ and improves the fixed-parameter tractable (FPT) time bounds for two graph classes. It presents a polynomial-time algorithm for tree skeletons and a faster FPT algorithm for chordal skeletons with runtime $O(n(2^{O(d^2k^2)}+n^2))$, improving over the prior $O(n(2^{O(d^4k^4)}+n^2))$ bound. The approach leverages clique-tree representations, LBFS orderings, and the Markov union concept to decompose counts across subgraphs via projections and extensions. These results significantly enhance practical MEC counting for chordal graphs and offer efficient trees-case computation, with potential impact on causal discovery and related graphical-model tasks.
Abstract
The structure of Markov equivalence classes (MECs) of causal DAGs has been studied extensively. A natural question in this regard is to algorithmically find the number of MECs with a given skeleton. Until recently, the known results for this problem were in the setting of very special graphs (such as paths, cycles, and star graphs). More recently, a fixed-parameter tractable (FPT) algorithm was given for this problem which, given an input graph $G$, counts the number of MECs with the skeleton $G$ in $O(n(2^{O(d^4k^4)} + n^2))$ time, where $n$, $d$, and $k$, respectively, are the numbers of nodes, the degree, and the treewidth of $G$. We give a faster FPT algorithm that solves the problem in $O(n(2^{O(d^2k^2)} + n^2))$ time when the input graph is chordal. Additionally, we show that the runtime can be further improved to polynomial time when the input graph $G$ is a tree.