A $\frac{4}{3}$-Approximation for the Maximum Leaf Spanning Arborescence Problem in DAGs

TL;DR

The paper addresses MLSA, the problem of finding a spanning arborescence rooted at $r$ with the maximum number of leaves in a DAG. It shows that MLSA in dags reduces naturally to a hereditary set packing problem, and further reduces to a bounded-set-size variant, enabling a $4/3$-approximation through a simple local-search algorithm that swaps up to $10$ sets. The core technical contribution is a two-stage weight-distribution analysis plus a structural argument that yields a $4/3$-approximation for the hereditary $3$-set packing problem, hence for MLSA in dags. It also establishes a lower bound indicating that, within constant-size local improvements, the $4/3$ ratio is tight, and discusses the potential for extending the approach to larger bounded sizes and other problems via set packing reductions.

Abstract

The Maximum Leaf Spanning Arborescence problem (MLSA) is defined as follows: Given a directed graph $G$ and a vertex $r\in V(G)$ from which every other vertex is reachable, find a spanning arborescence rooted at $r$ maximizing the number of leaves (vertices with out-degree zero). The MLSA has applications in broadcasting, where a message needs to be transferred from a source vertex to all other vertices along the arcs of an arborescence in a given network. In doing so, it is desirable to have as many vertices as possible that only need to receive, but not pass on messages since they are inherently cheaper to build. We study polynomial-time approximation algorithms for the MLSA. For general digraphs, the state-of-the-art is a $\min\{\sqrt{\mathrm{OPT}},92\}$-approximation. In the (still APX-hard) special case where the input graph is acyclic, the best known approximation guarantee of $\frac{7}{5}$ is due to Fernandes and Lintzmayer: They prove that any $α$-approximation for the \emph{hereditary $3$-set packing problem}, a special case of weighted $3$-set packing, yields a $\max\{\frac{4}{3},α\}$-approximation for the MLSA in acyclic digraphs (dags), and provide a $\frac{7}{5}$-approximation for the hereditary $3$-set packing problem. In this paper, we obtain a $\frac{4}{3}$-approximation for the hereditary $3$-set packing problem, and, thus, also for the MLSA in dags. In doing so, we manage to leverage the full potential of the reduction provided by Fernandes and Lintzmayer. The algorithm that we study is a simple local search procedure considering swaps of size up to $10$. Its analysis relies on a two-stage charging argument.

Figures (6)

• Figure 1: Illustration of the Maximum Leaf Spanning Arborescence problem. The leftmost picture shows a simple directed graph $G=(V,E)$, together with a vertex $r\in V$ from which every other vertex is reachable. The middle picture illustrates a spanning $r$-arborescence in $G$ with $3$ leaves (indicated by ). The rightmost picture shows a spanning $r$-arborescence in $G$ with $5$ leaves.
• Figure 2: The figure illustrates a spanning $r$-arborescence $T$(bold arcs) in a directed graph $G=(V,E)$ (bold and gray arcs). The non-leaf vertices are marked in different colors and for each non-leaf, the leaving arcs are drawn in the same color. Moreover, colorful frames indicate the out-neighborhoods of the non-leafs. It can be seen that these form a partition of $V\setminus \{r\}$. The number of leaves of $T$ can be calculated by summing up the colorful numbers written below the out-neighborhoods (cf. \ref{['prop:number_of_leaves']}).
• Figure 3: Construction of the conflict graph.
• Figure 4: The first step of the weight distribution.
• Figure 5: Illustration of the construction in the proof of Lemma \ref{['LemFirstStep']}. \ref{['SubFigLemSetConfiguration']} shows a collection $U\subseteq A$ of sets (blue, filled, solid), the collection $N(U,B_1\cup B_2)$ (red, dashed) of sets the sets in $U$ receive weight from in the first step, and further sets from $A$ (blue, not filled, solid) the sets in $N(U,B_1\cup B_2)$ send weight to. \ref{['SubFigLemX']} illustrates the construction of the set collection $X$.

Theorems & Definitions (54)

• Definition 1: Maximum Leaf Spanning Arborescence problem
• Definition 2: weighted $k$-set packing problem
• Definition 3: hereditary $3$-set packing problem
• Theorem 4: FernandesLintzmayer
• Definition 5: hereditary set packing problem
• Theorem 6
• Definition 7
• Proposition 8
• proof
• Proposition 9