FPT algorithms for packing $k$-safe spanning rooted sub(di)graphs

TL;DR

This paper develops fixed-parameter tractable algorithms for finding disjoint safe spanning subgraphs in graphs and digraphs, improving previous algorithms and answering open questions in the field.

Contribution

It introduces FPT algorithms for three related packing problems involving safe spanning structures, with new bounds and equivalences established.

Findings

Both arc-disjoint $k$-safe spanning arborescences and flow branchings are FPT with parameter $k$.

The problem of finding two arc-disjoint $(r,k)$-safe spanning trees is FPT with parameter $k$.

Existence of such structures is equivalent to bounded size and degree substructures.

Abstract

We study three problems introduced by Bang-Jensen and Yeo [Theor. Comput. Sci. 2015] and by Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016] about finding disjoint "balanced" spanning rooted substructures in graphs and digraphs, which generalize classic packing problems. Namely, given a positive integer $k$, a digraph $D=(V,A)$, and a root $r \in V$, we consider the problem of finding two arc-disjoint $k$-safe spanning $r$-arborescences and the problem of finding two arc-disjoint $(r,k)$-flow branchings. We show that both these problems are FPT with parameter $k$, improving on existing XP algorithms. The latter of these results answers a question of Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016]. Further, given an integer $k$, a graph $G=(V,E)$, and $r \in V$, we consider the problem of finding two arc-disjoint $(r,k)$-safe spanning trees. We show that this problem is also FPT with parameter $k$, again improving on a previous XP algorithm. Our main technical contribution is to prove that the existence of such spanning substructures is equivalent to the existence of substructures with size and maximum (out-)degree both bounded by a (linear or quadratic) function of $k$, which may be of independent interest.