Parameterized dynamic data structure for Split Completion
Konrad Majewski, Michał Pilipczuk, Anna Zych-Pawlewicz
TL;DR
The paper develops a randomized, parameterized dynamic data structure for Split Completion with fixed parameter $k$, achieving initialization time $k^{O(1)} d^2 n (\,\log n\,)^{O(1)}$ and amortized update time $5^{k} k^{O(1)} d^2 (\log n)^{O(1)}$, with correctness probability at least $1 - O(n^{-d})$. The approach combines a dynamic splittance maintenance core, promise-based edge listing across a split partition, a lifting wrapper for the general case, obstruction localization, and a branching framework that probabilistically resolves obstructions via edge insertions. A key methodological contribution is the integration of color-coding, sampling, and obstruction-based branching to simulate the static $5^{k}$-branching algorithm in a dynamic setting, yielding a Monte Carlo verification-friendly scheme. The work advances parameterized dynamic data structures for graph modification problems and suggests avenues for derandomization and tighter polylog improvements, with potential extension to related graph classes and problems. Overall, it demonstrates how to maintain yes/no answers for Split Completion under dynamic updates with provable, parameter-dependent efficiency.
Abstract
We design a randomized data structure that, for a fully dynamic graph $G$ updated by edge insertions and deletions and integers $k, d$ fixed upon initialization, maintains the answer to the Split Completion problem: whether one can add $k$ edges to $G$ to obtain a split graph. The data structure can be initialized on an edgeless $n$-vertex graph in time $n \cdot (k d \cdot \log n)^{\mathcal{O}(1)}$, and the amortized time complexity of an update is $5^k \cdot (k d \cdot \log n)^{\mathcal{O}(1)}$. The answer provided by the data structure is correct with probability $1-\mathcal{O}(n^{-d})$.