We develop a new algorithmic framework for designing approximation algorithms for cut-based optimization problems on capacitated undirected graphs that undergo edge insertions and deletions. Specifically, our framework dynamically maintains a variant of the hierarchical \(j\)-tree decomposition of [Madry FOCS’10], achieving a poly-logarithmic approximation factor to the graph’s cut structure and supporting edge updates in \(O(n^{\varepsilon})\) amortized update time, for any arbitrarily small constant \(\varepsilon \in (0,1)\).

We present a new solver-free parallel spectral sparsification algorithm for weighted graphs that relies only on parallel low-diameter decompositions and independent sampling. This yields the first algorithmic improvement over prior, solver-free parallel sparsification approaches since Koutis (2014) and, for the first time for a practical algorithm, eliminates any dependence on the target approximation accuracy ε in the algorithm's work and depth.

We give a simple algorithm for the dynamic approximate All-Pairs Shortest Paths (APSP) problem. Given a graph G = (V, E, l) with polynomially bounded edge lengths, our data structure processes |E| edge insertions and deletions in total time |E|^{1+o(1)} and provides query access to |E|^o(1)-approximate distances in time Õ(1) per query. We produce a data structure that mimics Thorup-Zwick distance oracles [Thorup and Zwick, 2005], but is dynamic and deterministic. Our algorithm selects a small number of pivot vertices. Then, for every other vertex, it reduces distance computation to maintaining distances to a small neighborhood around that vertex and to the nearest pivot. We maintain distances between pivots efficiently by representing them in a smaller graph and recursing. We maintain these smaller graphs by (a) reducing vertex count using the dynamic distance-preserving core graphs of Kyng-Meierhans-Probst Gutenberg [Kyng et al., 2024] in a black-box manner and (b) reducing edge-count using a dynamic spanner akin to Chen-Kyng-Liu-Meierhans-Probst Gutenberg [Chen et al., 2024]. Our dynamic spanner internally uses an APSP data structure. Choosing a large enough size reduction factor in the first step allows us to simultaneously bootstrap a spanner and a dynamic APSP data structure. Notably, our approach does not need expander graphs, an otherwise ubiquitous tool in derandomization.

We give a simple algorithm for maintaining a $n^{o(1)}$-approximate spanner $H$ of a graph $G$ with $n$ vertices as $G$ receives edge updates by reduction to the dynamic All-Pairs Shortest Paths (APSP) problem. Given an initially empty graph $G$, our algorithm processes $m$ insertions and $n$ deletions in total time $m^{1 + o(1)}$ and maintains an initially empty spanner $H$ with total recourse $n^{1 + o(1)}$. When the number of insertions is much larger than the number of deletions, this notably yields recourse sub-linear in the total number of updates. Our algorithm only has a single $O(\log n)$ factor overhead in runtime and approximation compared to the underlying APSP data structure. Therefore, future improvements for APSP will directly yield an improved dynamic spanner.