🤖 AI Summary
This paper addresses the problem of efficiently maintaining tree decompositions for dynamic graphs under edge insertions and deletions, specifically for graphs with treewidth at most $k$, aiming to sustain a tree decomposition of width at most $9k+8$ and support dynamic programming queries over it. The method introduces the novel concept of *downward well-linked* tree decompositions and integrates a splay-tree-inspired local rotation mechanism with amortized analysis. This yields the first fully dynamic algorithm achieving $O_k(log n)$ amortized update time—breaking the previous best $O_k(n^{o(1)})$ bound established at FOCS’23. Moreover, the approach enables a practical realization of the dynamic Courcelle’s theorem: it reduces the dependence on $n$ from $n^{o(1)}$ to $log n$, while delivering a simpler structural design and a more unified analytical framework.
📝 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 Soko{l}owski [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.