Dynamic Hierarchical $j$-Tree Decomposition and Its Applications

TL;DR

This work develops a dynamic framework for approximating cut-based graph optimization problems on capacitated undirected graphs under edge updates. It introduces a Dynamic Hierarchical $j$-tree Decomposition that maintains a small collection of $O(j)$-trees whose cores are of size $O(j)$ and that preserve the graph's cut structure up to a poly-logarithmic factor, with amortized update time $O(n^{\varepsilon})$ for any fixed $\varepsilon\in(0,1)$. A key technical ingredient is a fully dynamic cut sparsifier that handles vertex splits with very low recourse, enabling deep hierarchical maintenance without excessive recomputation. Leveraging these tools, the paper achieves poly-logarithmic approximations for all-pairs min-cut, sparsest cut, multi-way cut, and multi-cut in the fully dynamic setting, with truly sub-linear update and query times for several problems, representing a substantial advance over prior $n^{2/3}$-time barriers. The results have broad potential impact on practical dynamic graph algorithms and may influence future work on dynamic cut-based optimization and sparsification techniques.

Abstract

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 $O(n^ε)$ amortized update time, for any arbitrarily small constant $ε\in (0,1)$. Consequently, we obtain new trade-offs between approximation and update/query time for fundamental cut-based optimization problems in the fully dynamic setting, including all-pairs minimum cuts, sparsest cut, multi-way cut, and multi-cut. For the last three problems, these trade-offs give the first fully-dynamic algorithms achieving poly-logarithmic approximation in sub-linear time per operation. The main technical ingredient behind our dynamic hierarchy is a dynamic cut-sparsifier algorithm that can handle vertex splits with low recourse. This is achieved by white-boxing the dynamic cut sparsifier construction of [Abraham et al. FOCS'16], based on forest packing, together with new structural insights about the maintenance of these forests under vertex splits. Given the versatility of cut sparsification in both the static and dynamic graph algorithms literature, we believe this construction may be of independent interest.

Figures (4)

• Figure 4.1: Graphical depiction of how the core $C$ and forest $F$ might change after an update. Dotted edges denote edges that are in $G$ but not in $H$. The green edge denotes an edge newly inserted into $G$ and has to be processed by the data structure. The orange edges denote the edges $e_{\min}^F(u)$ and $e_{\min}^F(v)$. In (b), the vertices $u,v$ are added as roots to the set $R$. In (c), the minimum capacity edges have been removed from the forest and been added as projected edges into the core. Moreover, two projected edges in the core have their endpoints moved from $w$ to $u$.
• Figure 6.1: Summary of initialization.
• Figure 6.2: Summary of update.
• Figure 7.1: Graphical depiction of \ref{['lma:max-flow-struct']} and how our algorithm finds an (approximate) $u$-$v$-min-cut of a $j$-tree. $S_1, S_3$ correspond to the minimum cuts separating $u,v$ respectively from their roots in $F$. Then, $S_2$ denotes the minium cut between $\mathsf{root}_F(u)$ and $\mathsf{root}_F(v)$ in the core $C$. The minimum $u$-$v$ cut corresponds to the minimum capacity cut of these three candidates.

Theorems & Definitions (137)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Definition 2.3: Vertex Split
• Theorem 2.4
• Lemma 2.5
• Definition 3.1: Spanning Forest
• Definition 3.2: Rooted Forest
• Definition 3.3: Decremental Rooted Forest