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Dynamic Treewidth in Logarithmic Time

Tuukka Korhonen

TL;DR

This work delivers a dynamic data structure that maintains a tree decomposition of a dynamic graph with width at most $9\cdot\mathsf{tw}(G)+8$ while supporting edge insertions and deletions with amortized time $2^{O(k)}\log n$, where $k$ is an upper bound on treewidth. The key innovation is a downwardly well-linked, or downwards well-linked, superbranch decomposition that supports local rotations and contractions, enabling splay-tree-like amortized analysis. It also shows how to maintain arbitrary dynamic programming schemes on the decomposition via tree-decomposition automata and prefix-rebuilding updates, yielding dynamic Courcelle-type results with $O_k(\log n)$ amortized update time. The approach improves prior subpolynomial-time dynamic treewidth structures and offers a simpler, modular framework with broad potential applications in dynamic graph algorithms and parameterized DP. Overall, the paper advances practical dynamic graph algorithms by achieving near-constant-exponential dependence on $k$ in update time while maintaining strong decomposition guarantees.

Abstract

We present a dynamic data structure that maintains a tree decomposition of width at most $9k+8$ of a dynamic graph with treewidth at most $k$, which is updated by edge insertions and deletions. The amortized update time of our data structure is $2^{O(k)} \log n$, where $n$ is the number of vertices. The data structure also supports maintaining any ``dynamic programming scheme'' on the tree decomposition, providing, for example, a dynamic version of Courcelle's theorem with $O_{k}(\log n)$ amortized update time; the $O_{k}(\cdot)$ notation hides factors that depend on $k$. This improves upon a result of Korhonen, Majewski, Nadara, Pilipczuk, and Sokołowski [FOCS 2023], who gave a similar data structure but with amortized update time $2^{k^{O(1)}} n^{o(1)}$. Furthermore, our data structure is arguably simpler. Our main novel idea is to maintain a tree decomposition that is ``downwards well-linked'', which allows us to implement local rotations and analysis similar to those for splay trees.

Dynamic Treewidth in Logarithmic Time

TL;DR

This work delivers a dynamic data structure that maintains a tree decomposition of a dynamic graph with width at most while supporting edge insertions and deletions with amortized time , where is an upper bound on treewidth. The key innovation is a downwardly well-linked, or downwards well-linked, superbranch decomposition that supports local rotations and contractions, enabling splay-tree-like amortized analysis. It also shows how to maintain arbitrary dynamic programming schemes on the decomposition via tree-decomposition automata and prefix-rebuilding updates, yielding dynamic Courcelle-type results with amortized update time. The approach improves prior subpolynomial-time dynamic treewidth structures and offers a simpler, modular framework with broad potential applications in dynamic graph algorithms and parameterized DP. Overall, the paper advances practical dynamic graph algorithms by achieving near-constant-exponential dependence on in update time while maintaining strong decomposition guarantees.

Abstract

We present a dynamic data structure that maintains a tree decomposition of width at most of a dynamic graph with treewidth at most , which is updated by edge insertions and deletions. The amortized update time of our data structure is , where is the number of vertices. The data structure also supports maintaining any ``dynamic programming scheme'' on the tree decomposition, providing, for example, a dynamic version of Courcelle's theorem with amortized update time; the notation hides factors that depend on . This improves upon a result of Korhonen, Majewski, Nadara, Pilipczuk, and Sokołowski [FOCS 2023], who gave a similar data structure but with amortized update time . Furthermore, our data structure is arguably simpler. Our main novel idea is to maintain a tree decomposition that is ``downwards well-linked'', which allows us to implement local rotations and analysis similar to those for splay trees.

Paper Structure

This paper contains 60 sections, 37 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Computing treewidth.
  3. Dynamic treewidth.
  4. Dynamic treewidth $1$.
  5. Dynamic treewidth $2$ and $3$.
  6. Dynamic treewidth $k$.
  7. Our contribution.
  8. Applications.
  9. Our techniques.
  10. Organization.
  11. Outline
  12. Downwards well-linked superbranch decompositions.
  13. Operations on downwards well-linked superbranch decompositions.
  14. Balancing.
  15. Inserting and deleting edges.
  16. ...and 45 more sections

Key Result

Theorem 1.0

There is a data structure that is initialized with an edgeless $n$-vertex graph $G$ and an integer $k$, supports updating $G$ via edge insertions and deletions under the promise that $\mathsf{tw}(G) \le k$ at all times, and maintains a rooted tree decomposition of $G$ of width at most $9 \cdot \math

Figures (2)

  • Figure 1: Splitting a node $t$ of a superbranch decomposition $\mathcal{T}$ using a well-linked set $A = \{e_a, e_b\}$ of $\mathsf{torso}(t)$, which corresponds to a well-linked set $A \triangleright \mathcal{T}$ of $G$.
  • Figure 2: The balancing subroutine.

Theorems & Definitions (71)

  • Theorem 1.0
  • Corollary 1.0
  • Corollary 1.0
  • Lemma 2.1: Informal version of \ref{['lem:wltransindecomp']}
  • Lemma 2.2: Corresponds to \ref{['lem:partitiontowlalg']}
  • Lemma 3.0: RobertsonS-GMXIII,
  • Lemma 4.0: RobertsonS-GMXIII,
  • Lemma 4.1: DBLP:journals/corr/abs-2411-02658
  • proof
  • Lemma 4.2
  • ...and 61 more