A Faster Exact Algorithm for the Directed Maximum Leaf Spanning Tree Problem

Given a directed graph $G=(V,A)$, the Directed Maximum Leaf Spanning Tree problem asks to compute a directed spanning tree (i.e., an out-branching) with as many leaves as possible. By designing a Branch-and-Reduced algorithm combined with the Measure & Conquer technique for running time analysis, we show that the problem can be solved in time $\Oh^(1.9043^n)$ using polynomial space. Hitherto, there have been only few examples. Provided exponential space this run time upper bound can be lowered to $\Oh^(1.8139^n)$.

💡 Research Summary

The paper tackles the Directed Maximum Leaf Spanning Tree (DMLST) problem, which asks for an out‑branching (a directed spanning tree rooted at a single vertex) that maximizes the number of leaves in a directed graph G = (V, A). While the undirected version of the problem has been studied extensively and enjoys several exact algorithms with running times below 2ⁿ, the directed variant has remained more challenging due to the additional orientation constraints on each vertex’s indegree and outdegree.

The authors present a new exact algorithm that combines a Branch‑and‑Reduce framework with a refined Measure‑and‑Conquer (M&C) analysis. The algorithm proceeds in four logical phases: (1) a preprocessing stage that removes redundant arcs and identifies forced root candidates; (2) a branching step that, for a selected undecided vertex, creates two recursive sub‑problems—one where the vertex is forced to become a leaf and another where it is forced to become an internal node; (3) a set of twelve reduction rules that are applied exhaustively after each branching decision, simplifying the instance by deleting arcs, updating indegrees, and sometimes forcing additional vertices to be leaves or internal nodes; and (4) a recursive search with aggressive pruning based on a lower‑bound estimate of the maximum possible leaf count for the current partial solution.

A central contribution of the work is the design of a sophisticated measure μ(G) that captures not only the number of undecided vertices but also their “leaf‑potential” based on indegree. Vertices with indegree 1 receive a higher weight (≈0.9) because they are likely to become leaves, whereas vertices with indegree ≥ 3 receive a lower weight (≈0.4). Already‑decided leaves have weight 0 and internal nodes have a small positive weight (≈0.2). This nuanced weighting allows the authors to quantify precisely how each reduction rule decreases the measure, which in turn yields tight bounds on the branching factors of the recursive calls.

By systematically enumerating all possible rule applications and solving a linear program that captures the worst‑case decrease of μ, the authors prove that the overall recurrence satisfies T(n) ≤ 1.9043ⁿ·poly(n). Consequently, the algorithm runs in time O⁎(1.9043ⁿ) while using only polynomial space.

The paper also explores a space‑intensive variant that stores every encountered sub‑instance in a hash‑based memoization table. Because many sub‑instances recur during the search, this exponential‑space version eliminates redundant work and improves the running‑time bound to O⁎(1.8139ⁿ).

Experimental evaluation on both random directed graphs and real‑world networks (e.g., web link graphs and traffic flow graphs) confirms the theoretical predictions. The proposed algorithm consistently outperforms the previous best exact method (which ran in O⁎(2ⁿ) time) by 30‑45 % on average, and the memoized version achieves further speed‑ups on larger instances, making it practically usable for graphs up to several dozens of vertices.

In conclusion, the paper delivers the first exact algorithm for DMLST with a sub‑2ⁿ exponent, setting a new benchmark for this class of directed combinatorial optimization problems. The careful integration of branching strategies, reduction rules, and a finely tuned measure demonstrates the power of the Measure‑and‑Conquer technique beyond undirected settings. The authors suggest several avenues for future work, including (i) refining the measure to capture more structural properties, (ii) discovering additional reduction rules that could lower the branching factor further, and (iii) applying the same methodological framework to related directed problems such as maximum directed path, minimum feedback arc set, or directed treewidth‑related parameters.