Additive One Approximation for Minimum Degree Spanning Tree: Breaking the $O(mn)$ Time Barrier

TL;DR

A deterministic algorithm is provided that returns an additive one approximate optimal spanning tree in $\tilde{O}(mn^{3/4})$ time that constitutes a substantive progress towards answering an open question that has been repeatedly posed in the literature.

Abstract

We consider the ``minimum degree spanning tree'' problem. As input, we receive an undirected, connected graph $G=(V, E)$ with $n$ nodes and $m$ edges, and our task is to find a spanning tree $T$ of $G$ that minimizes $\max_{u \in V} \text{deg}_T(u)$, where $\text{deg}_T(u)$ denotes the degree of $u \in V$ in $T$. The problem is known to be NP-hard. In the early 1990s, an influential work by Fürer and Raghavachari presented a local search algorithm that runs in $\tilde{O}(mn)$ time, and returns a spanning tree with maximum degree at most $Δ^\star+1$, where $Δ^\star$ is the optimal objective. This remained the state-of-the-art runtime bound for computing an additive one approximation, until now. We break this $O(mn)$ runtime barrier dating back to three decades, by providing a deterministic algorithm that returns an additive one approximate optimal spanning tree in $\tilde{O}(mn^{3/4})$ time. This constitutes a substantive progress towards answering an open question that has been repeatedly posed in the literature [Pettie'2016, Duan and Pettie'2020, Saranurak'2024]. Our algorithm is based on a novel application of the blocking flow paradigm.

Figures (6)

• Figure 1: An example of a sub-tree $T^\mathcal{F}_{u\leftarrow v}$.
• Figure 2: An example of a molecule and its atoms. Solid line are forest edges, ans dashed lines are non-forest edges. The value of $\Delta^\star$ is $3$. Red nodes correspond to non-reducible nodes. Yellow nodes are reducible nodes of degree $\Delta^\star+1$. Green nodes are reducible nodes of degree $\leq \Delta^\star$. Atoms are depicted with magenta curves. (a) The status of the initial molecule. (b) The status of the molecule after reducing the degree of $u$ in atom $3$. The sub-tree inside atom $3$ will change but all of the other atoms and non-reducible nodes remain unaffected.
• Figure 3: An example of an augmenting chain. The value of $\Delta^\star$ is $3$. $\mathcal{M}$-molecules are depicted with blue ovals. Green and Yellow nodes are $\mathcal{M}$-reducible nodes of degree $\leq \Delta^\star$ and $=\Delta^\star+1$, respectively. Red nodes are $\mathcal{M}$-non-reducible. Black nodes are $\mathcal{M}$-free. $\mathcal{M}$-blocks of the augmenting chain are depicted by red curves.
• Figure 4: The result of applying augmenting chain in \ref{['fig:aug-chain']}. Molecules $1, 2, 4, 6,$ and $9$ are affected and their nodes become free. Molecules $3,5,7,$ and $8$ remain unaffected. The node $d$ is dirty according to the definition.
• Figure 5: An example of a $\theta$-molecular decomposition of some forest $\mathcal{F}$ for $\theta = 4$.

Theorems & Definitions (67)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1: FR92
• proof
• Lemma 3.1: FR92
• proof
• Lemma 3.3
• Definition 3.4
• Lemma 3.5
• Lemma 3.6