A Simple 2-Approximation for Maximum-Leaf Spanning Tree

TL;DR

This work addresses the MAX leaf spanning tree problem on a connected simple graph, which is MAXSNP-complete and NP-hard under planar and bounded-degree restrictions. It presents a simple $O(m)$-time 2-approximation algorithm, tree$(G)$, that greedily expands a growing tree by attaching all outside neighbors of a selected leaf, with a priority order among leaves given by $W_2(T)$, $W_1(T)$, and $W_0(T)$ and a one-round lookahead when leaf expansion stalls. A rank-based analysis using a rank function $r(v)$ yields a concise proof of the 2-approximation and demonstrates tightness via a constructed graph. The linear-time implementation relies on basic data structures, achieving the claimed $O(m)$ running time and offering a simpler alternative to prior multi-phase approaches. Overall, the paper advances efficient approximations for MAXSNP-complete problems and motivates further exploration of starting-vertex choices to potentially improve the constant factor.

Abstract

For an $m$-edge connected simple graph $G$, finding a spanning tree of $G$ with the maximum number of leaves is MAXSNP-complete. The problem remains NP-complete even if $G$ is planar and the maximal degree of $G$ is at most four. Lu and Ravi gave the first known polynomial-time approximation algorithms with approximation factors $5$ and $3$. Later, they obtained a $3$-approximation algorithm that runs in near-linear time. The best known result is Solis-Oba, Bonsma, and Lowski's $O(m)$-time $2$-approximation algorithm. We show an alternative simple $O(m)$-time $2$-approximation algorithm whose analysis is simpler. This paper is dedicated to the cherished memory of our dear friend, Professor Takao Nishizeki.

Figures (4)

• Figure 1: Our algorithm on a connected graph $G$.
• Figure 2: An example of the spanning tree $T$ returned by our algorithm. The edges of $T$ are represented by the solid lines. (a) Vertex $i$ is the $i$-th expanded vertex. The vertices with no labels are the leaves of the returned spanning tree. (b) The label of each vertex is its rank according to the expansion order given in (a).
• Figure 3: The solid edges form a spanning forest $F$ having $6$ connected components, three of which have at least three vertices. Any two vertices are connected in $F$ if and only if their ranks (as shown in Figure \ref{['figure:figure2']}(b)) are identical. The dark vertex in each connected component $F_i$ with $1\leq i\leq 6$ is a nonleaf vertex $f_i$ of $F_i$. (a) The $15$ dotted edges form a spanning tree $R$. The $11$ dark dotted edges form the minimal subtree $R'$ of $R$ that spans the dark vertices $f_1,\ldots,f_6$. (b) The $5$ dotted edges form a minimal forest $S$ with $|V(S)|=9$ of $R'$ such that $F\cup S$ is a spanning tree. $S$ has four connected components.
• Figure 4: The spanning tree $T$ consists of the solid edges. The number of each internal vertex of $T$ represents its expansion order.

Theorems & Definitions (9)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof