[Paper Review] Better Algorithms and Bounds for Directed Maximum Leaf Problems

Noga Alon, Fedor V. Fomin+3 more|ArXiv.org|Jul 7, 2007

Advanced Graph Theory Research12 references4 citations

TL;DR

This paper presents improved algorithms and combinatorial bounds for the Directed Maximum Leaf Out-Branching problem in strongly connected digraphs. By leveraging local search and pathwidth-based structural analysis, the authors achieve a runtime of $2^{O(k\log^2 k)} \cdot n^{O(1)}$ for detecting out-branchings with at least $k$ leaves, and prove that every strongly connected digraph with minimum in-degree 3 contains an out-branching with at least $(n/4)^{1/3} - 1$ leaves.

ABSTRACT

The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that \begin{itemize} \item every strongly connected digraph $D$ of order $n$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; \item if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying graph is $O(k\log k)$; \item it can be decided in time $2^{O(k\log^2 k)}\cdot n^{O(1)}$ whether a strongly connected digraph on $n$ vertices has an out-branching with at least $k$ leaves. \end{itemize} All improvements use properties of extremal structures obtained after applying local search and of some out-branching decompositions.

Motivation & Objective

• To develop faster fixed-parameter tractable algorithms for the Directed Maximum Leaf Out-Branching problem in strongly connected digraphs.
• To establish tighter lower bounds on the number of leaves in out-branchings of strongly connected digraphs with minimum in-degree 3.
• To explore the relationship between the number of leaves in an out-branching and the pathwidth of the underlying undirected graph.
• To extend algorithmic results to broader classes of digraphs, including acyclic and general digraphs, via structural decomposition.

Proposed method

• Employing local search techniques to iteratively improve candidate out-branchings by reconfiguring subtrees to increase leaf count.
• Using a novel tree decomposition approach based on the $\beta$-decomposition of out-branchings to bound structural complexity.
• Applying pathwidth analysis to the underlying undirected graph of a digraph to derive parameterized runtime bounds.
• Establishing a recursive path decomposition argument that bounds the pathwidth of subgraphs induced by nodes at each layer of a tree decomposition.
• Leveraging the property that if no out-branching with $k$ leaves exists, then the underlying graph has pathwidth $O(k\log k)$, to guide algorithm design.
• Extending results from strongly connected digraphs to acyclic digraphs and general digraphs using the $\ell(D) = \max_v \ell_s(D_v)$ identity, where $D_v$ is the subdigraph reachable from vertex $v$.

Experimental results

Research questions

• RQ1 What is the best possible lower bound on the number of leaves in an out-branching of a strongly connected digraph with minimum in-degree 3?
• RQ2 Can the parameterized complexity of the Directed Maximum Leaf Out-Branching problem be improved beyond previous exponential dependencies on $k$?
• RQ3 How does the pathwidth of the underlying undirected graph relate to the existence of an out-branching with $k$ leaves?
• RQ4 Can the algorithmic techniques developed for strongly connected digraphs be extended to general digraphs?
• RQ5 Is the $\Omega(n^{1/3})$ lower bound on the number of leaves tight, or can it be improved to $\Omega(n^r)$ for $r > 1/3$?

Key findings

• Every strongly connected digraph of order $n$ with minimum in-degree at least 3 contains an out-branching with at least $(n/4)^{1/3} - 1$ leaves.
• If a strongly connected digraph does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying undirected graph is $O(k\log k)$.
• The $k$-DMLOB problem on strongly connected digraphs can be solved in time $2^{O(k\log^2 k)} \cdot n^{O(1)}$, improving upon the previous bound of $2^{O(k^2\log k)} \cdot n^{O(1)}$.
• For acyclic digraphs, the problem can be solved in $2^{O(k\log k)} \cdot n^{O(1)}$ time, which is faster than the strongly connected case.
• The algorithmic results extend to general digraphs for the $k$-DMLOT problem (maximum leaf out-tree), with the same asymptotic runtime.
• There exist strongly connected digraphs with minimum in-degree 3 in which every out-branching has at most $O(\sqrt{n})$ leaves, showing that the $n^{1/3}$ bound is not tight in the worst case.