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

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 O(λ14.5√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(λ³ 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 O(λ5.5√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 O(m1-1/12) amortized update time, improving upon O(m1-1/31) [Goranci et al., SODA 2023]. We also give the first fully dynamic algorithm that maintains a (1+ε)-approximation of the fractional arboricity -- which is strictly harder than the integral arboricity. Our algorithm is deterministic and has O(α log⁶m/ε⁴) amortized update time, for arboricity at most α. We extend these results to a Monte Carlo algorithm with O(poly(log m,ε⁻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].