Tree-Packing Revisited: Faster Fully Dynamic Min-Cut and Arboricity

TL;DR

This work rethinks tree-packing as a central tool for dynamic graph problems by tying it more tightly to min-cut and fractional arboricity. It proves that a far smaller greedy-tree packing, on the order of $\Theta(\lambda^3\log m)$ trees, suffices to certify either a 1-respecting cut or a trivial cut in a contracted graph, enabling faster fully dynamic min-cut algorithms. The authors present a deterministic exact dynamic min-cut with update time $\tilde{O}(\lambda_{max}^{5.5}\sqrt{n})$ for small $\lambda$, and a general amortized bound of $\tilde{O}(m^{1-1/12})$, as well as efficient multi-graph extensions; they also introduce the first dynamic $(1+\varepsilon)$-approximation for fractional arboricity with deterministic guarantees and extend to Monte Carlo methods against adaptive adversaries. A key structural insight links arboricity to the ideal load decomposition, yielding both theoretical lower bounds for greedy packing and constructive small-packings that achieve near-optimal load approximation. Together, these results advance dynamic graph algorithms by reducing reliance on large tree-packings and by unifying min-cut and arboricity through a refined tree-packing framework with practical update-time improvements.

Abstract

A tree-packing is a collection of spanning trees of a graph. It has been a useful tool for computing the minimum cut in static, dynamic, and distributed settings. In particular, [Thorup, Comb. 2007] used them to obtain his dynamic min-cut algorithm with $\tilde O(λ^{14.5}\sqrt{n})$ worst-case update time. We reexamine this relationship, showing that we need to maintain fewer spanning trees for such a result; we show that we only need to pack $Θ(λ^3 \log m)$ greedy trees to guarantee a 1-respecting cut or a trivial cut in some contracted graph. Based on this structural result, we then provide a deterministic algorithm for fully dynamic exact min-cut, that has $\tilde O(λ^{5.5}\sqrt{n})$ worst-case update time, for min-cut value bounded by $λ$. In particular, this also leads to an algorithm for general fully dynamic exact min-cut with $\tilde O(m^{1-1/12})$ amortized update time, improving upon $\tilde O(m^{1-1/31})$ [Goranci et al., SODA 2023]. We also give the first fully dynamic algorithm that maintains a $(1+\varepsilon)$-approximation of the fractional arboricity -- which is strictly harder than the integral arboricity. Our algorithm is deterministic and has $O(α\log^6m/\varepsilon^4)$ amortized update time, for arboricity at most $α$. We extend these results to a Monte Carlo algorithm with $O(\text{poly}(\log m,\varepsilon^{-1}))$ amortized update time against an adaptive adversary. Our algorithms work on multi-graphs as well. Both result are obtained by exploring the connection between the min-cut/arboricity and (greedy) tree-packing. We investigate tree-packing in a broader sense; including a lower bound for greedy tree-packing, which - to the best of our knowledge - is the first progress on this topic since [Thorup, Comb. 2007].

Figures (5)

• Figure 1: The graph with two trees of the tree-packing colored. Observe that all edges are packed once in these two trees except for $a$, which is packed twice, and $b$ which is never packed.
• Figure 2: Various stages of the construction of the graph $G_{n,k}$. The leftmost illustration shows $K_3$ with an additional parallel edge. The middle illustration shows the graph obtained by repeatedly applying the operation to the graph to the left. Finally, the rightmost illustration shows the final graph $G_{n,k}$.
• Figure 3: The construction with the level $Y_2$ highlighted in orange.
• Figure 4: The figure illustrates $T$ (on the left) and $T'$ (on the right) in the case where $i = 1$, $s = 3$, $\text{lev}(Y_1) = \text{lev}(Y_2) = \text{lev}(Y_3) = 2$, and $\text{lev}(Y_4) = 1$.
• Figure 5: The figure illustrates $T_1$ (on the left) and $T'_1$ (on the right) in the case where $i = 1$, $s = 3$, $\text{lev}(Y_1) = \text{lev}(Y_2) = \text{lev}(Y_3) = 2$, and $\text{lev}(Y_4) = 1$. Observe that all edges are packed once except for $a$, which is packed twice, and $b$ which is not packed.

Theorems & Definitions (70)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 8
• Theorem 8