A fixed-parameter algorithm for dominance drawings of DAGs

A weak dominance drawing Γ of a DAG $G=(V,E)$ is a d-dimensional drawing such that $D\left(u\right)<D\left(v\right)$ for every dimension D of Γ if there is a directed path from a vertex u to a vertex v in G, where $D\left(w\right)$ is the coordinate of vertex $w\in V$ in dimension D of Γ. If $D\left(u\right)<D\left(v\right)$ for every dimension D of Γ, but there is no path from u to v, we have a falsely implied path (fip). Minimizing the number of fips is an important theoretical and practical problem. Computing 2-dimensional weak dominance drawings with minimum number of fips is NP-hard. We show that this problem is FPT parameterized by the dimension d and the modular width mw. A key ingredient of our proof is the Compaction Lemma, where we show an interesting property of any weak dominance drawing of G with the minimum number of fips. This FPT result in weak dominance, which is interesting by itself because the fip-minimization problem is NP-hard, is used to prove our main contributions. Computing the dominance dimension of G, that is, the minimum number of dimensions d for which G has a d-dimensional dominance drawing (a weak dominance drawing with 0 fips), is a well-known NP-hard problem. We show that the dominance dimension of G is bounded by $\frac{mw}{2}$ (or mw, if $mw<4$) and that computing the dominance dimension of G is an FPT problem with parameter mw. As far as we know, this the first FPT-algorithm to compute the dominance dimension of a DAG.