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 \(\tilde O(\lambda^{14.5}\sqrt{n})\) 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 \(\Theta(\lambda^3 \log m)\) 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 \(\tilde O(\lambda^{5.5}\sqrt{n})\) worst-case update time, for min-cut value bounded by \(\lambda\). 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 \(O(\alpha \log^6m/\varepsilon^4)\) amortized update time, for arboricity at most \(\alpha\). 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].