[Paper Review] Tight Bounds and Faster Algorithms for Directed Max-Leaf Problems

Paul Bonsma, Frederic Dorn|ArXiv.org|Apr 12, 2008

Advanced Graph Theory Research21 references3 citations

TL;DR

This paper presents fixed-parameter tractable (FPT) algorithms for the directed Max-Leaf Out-Tree and Max-Leaf Out-Branching problems, achieving a time complexity of $2^{O(k\log k)} \cdot n^{O(1)}$, which improves upon previous bounds. The key contribution is a combinatorial bound showing that for digraphs without useless arcs, the maximum number of leaves in an out-branching is at least one-third of the maximum in any out-tree, enabling efficient algorithm design.

ABSTRACT

An out-tree $T$ of a directed graph $D$ is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By $l(D)$ and $l_s(D)$ we denote the maximum number of leaves over all out-trees and out-branchings of $D$, respectively. We give fixed parameter tractable algorithms for deciding whether $l_s(D)\geq k$ and whether $l(D)\geq k$ for a digraph $D$ on $n$ vertices, both with time complexity $2^{O(k\log k)} \cdot n^{O(1)}$. This improves on previous algorithms with complexity $2^{O(k^3\log k)} \cdot n^{O(1)}$ and $2^{O(k\log^2 k)} \cdot n^{O(1)}$, respectively. To obtain the complexity bound in the case of out-branchings, we prove that when all arcs of $D$ are part of at least one out-branching, $l_s(D)\geq l(D)/3$. The second bound we prove in this paper states that for strongly connected digraphs $D$ with minimum in-degree 3, $l_s(D)\geq Θ(\sqrt{n})$, where previously $l_s(D)\geq Θ(\sqrt[3]{n})$ was the best known bound. This bound is tight, and also holds for the larger class of digraphs with minimum in-degree 3 in which every arc is part of at least one out-branching.

Motivation & Objective

• To develop faster fixed-parameter tractable algorithms for the $k$-Leaf Out-Tree and $k$-Leaf Out-Branching problems in directed graphs.
• To improve upon existing FPT time complexities, particularly reducing the parameter function from $2^{O(k^3\log k)}$ to $2^{O(k\log k)}$.
• To establish tight combinatorial lower bounds on the number of leaves in out-branchings of strongly connected digraphs with minimum in-degree 3.
• To analyze the ratio $\ell(D)/\ell_s(D)$ and show it is at most 3 for digraphs without useless arcs, improving algorithmic efficiency.

Proposed method

• Design a dynamic programming-based FPT algorithm using tree decomposition and vertex coloring techniques to explore out-tree structures.
• Introduce a mapping from vertices in a path to leaves in a tree to bound the number of preimages per leaf, ensuring at most three preimages per leaf.
• Prove that in digraphs without useless arcs, $\ell_s(D) \geq \ell(D)/3$, enabling reduction of the out-branching problem to the out-tree problem with bounded loss.
• Use a staged construction of a path $P$ in a tree $T_p$ to show that at least $p/4$ vertices of $P$ become leaves, leading to a lower bound on the number of leaves.
• Apply the green/white/leaf vertex coloring scheme to track structural changes during algorithmic stages and ensure correctness of the leaf counting.
• Leverage the fact that in acyclic or strongly connected digraphs, out-trees with $k$ leaves can be extended to out-branchings, allowing shared algorithmic frameworks.

Experimental results

Research questions

• RQ1 Can the parameter function of FPT algorithms for $k$-Leaf Out-Branching be improved beyond $2^{O(k^3\log k)}$?
• RQ2 What is the tightest possible upper bound on the ratio $\ell(D)/\ell_s(D)$ for directed graphs without useless arcs?
• RQ3 Is it possible to achieve a $2^{O(k)}$-time FPT algorithm for directed max-leaf problems, similar to the undirected case with $c^k$ for $c=6.75$?
• RQ4 What is the best possible lower bound on $\ell_s(D)$ for strongly connected digraphs with minimum in-degree 3?
• RQ5 Can the factor $1/4$ in the lower bound for the number of leaves in an out-branching be improved?

Key findings

• The paper achieves an FPT algorithm for $k$-Leaf Out-Tree with time complexity $2^{O(k\log k)} \cdot n^{O(1)}$, improving over the prior $2^{O(k\log^2 k)}$ bound.
• For $k$-Leaf Out-Branching, the algorithm runs in $2^{O(k\log k)} \cdot n^{O(1)}$ time, improving upon the previous $2^{O(k^3\log k)}$ complexity.
• It is proven that $\ell_s(D) \geq \ell(D)/3$ for all digraphs without useless arcs, providing a crucial structural link between out-trees and out-branchings.
• For strongly connected digraphs with minimum in-degree 3, $\ell_s(D) \geq \Theta(\sqrt{n})$, which is tight and improves the prior $\Theta(\sqrt[3]{n})$ bound.
• The bound $\ell_s(D) \geq \Theta(\sqrt{n})$ also holds for digraphs with minimum in-degree 3 where every arc is part of at least one out-branching.
• The analysis shows that every leaf in the constructed tree has at most three preimages in the vertex path, which is used to derive the $p/4$ lower bound on the number of leaves.