Tight Bounds and Faster Algorithms for Directed Max-Leaf Problems

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, $l_s(D)\geq \Theta(\sqrt{n})$, where previously $l_s(D)\geq \Theta(\sqrt[3]{n})$ 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.

💡 Research Summary

This paper studies two closely related optimization problems on directed graphs: finding an out‑tree (a rooted arborescence that need not span all vertices) with the maximum possible number of leaves, and finding an out‑branching (a spanning out‑tree) with the maximum number of leaves. The quantities ℓ(D) and ℓ_s(D) denote, respectively, the maximum leaf count over all out‑trees and over all out‑branchings of a digraph D. The decision versions—k‑Leaf Out‑Tree (does ℓ(D) ≥ k?) and k‑Leaf Out‑Branching (does ℓ_s(D) ≥ k?)—are NP‑hard, as they generalize the classic undirected k‑Leaf Spanning Tree problem.

Algorithmic contribution.
Previous fixed‑parameter tractable (FPT) results achieved running times of 2^{O(k log² k)}·n^{O(1)} for k‑Leaf Out‑Tree and 2^{O(k³ log k)}·n^{O(1)} for k‑Leaf Out‑Branching. The authors present new algorithms for both problems with a unified parameter function f(k)=2^{O(k log k)}. The key ideas are:

1. Removal of useless arcs. An arc is “useless” if it never belongs to any out‑branching. Such arcs can be identified and deleted in quadratic time, leaving a digraph where every remaining arc participates in at least one out‑branching.

2. Construction of a 1‑optimal out‑branching. Starting from an arbitrary root r in each strong component, a locally optimal out‑branching T (no single‑arc exchange can increase the leaf count) is computed in polynomial time.

3. Tree‑decomposition based on T. Instead of the path‑decomposition used in earlier work (which yielded width O(k³)), the authors use T itself as the backbone of a tree‑decomposition. For each vertex v of T, the corresponding bag X_v contains v, its parent, all branch successors, all leaves, and the heads of back‑arcs incident to v. This construction guarantees width at most 4k−5.

4. Dynamic programming on the decomposition. With width O(k), the DP explores all possible leaf/branch configurations inside each bag. The number of DP states per bag is 2^{O(k log k)}, leading to the overall running time 2^{O(k log k)}·n^{O(1)}.

The algorithm first checks two easy win conditions: (a) if the 1‑optimal T already has ≥k leaves, or (b) if some vertex z has at least k back‑arcs whose heads are distinct. In either case the answer is YES. Otherwise the tree‑decomposition and DP are invoked.

Combinatorial bounds.
Two structural results underpin the algorithmic improvements:

• Ratio bound ℓ(D)/ℓ_s(D) ≤ 3. For digraphs without useless arcs, the authors prove that any out‑tree can be transformed into an out‑branching losing at most a factor‑3 in leaf count. The proof analyses the relationship between leaves, branch vertices, and back‑arcs in a 1‑optimal out‑branching, showing that either enough back‑arcs exist to directly augment leaves, or the DP will succeed. This constant‑factor bound immediately yields an FPT algorithm for k‑Leaf Out‑Branching with the same f(k) as for the out‑tree case.

• Lower bound ℓ_s(D) = Ω(√n) for minimum in‑degree 3. Earlier work only guaranteed ℓ_s(D) ≥ Ω(n^{1/3}) for strongly connected digraphs with δ⁻(D) ≥ 3. By refining the argument from