A Note on Rounding Matchings in General Graphs
Aditi Dudeja
TL;DR
The paper extends Wajc's fully adaptive rounding of dynamic fractional matchings from bipartite to general graphs, preserving near-optimal value with polylogarithmic update time. It introduces a weak-degree sparsifier via Sparsification and analyzes entropy-regularized matching tailored to the weaker sparsifier, establishing weight preservation under adaptivity and strong duality. The authors then show how to derive a decremental, fully adaptive $(1-\varepsilon)$-approximate maximum weight matching with amortized time $\tilde{O}(\varepsilon^{-41} + \frac{n^2}{m}\varepsilon^{-6})$, leveraging the sparsifier and entropy-regularized solutions. Together, these results broaden the applicability of dynamic rounding to general graphs and improve robustness against adaptive adversaries in dynamic weighted matching problems.
Abstract
In this note, we revisit the rounding algorithm of Wajc. Wajc gave a fully-adaptive randomized algorithm that rounds a dynamic fractional matching in an unweighted bipartite graph to an integral matching of nearly the same value in $O(\text{poly}(\log n,\frac{1}{\varepsilon}))$ update time. We give show that the guarantees of this algorithm hold for general graphs as well. Additionally, we show useful properties of this subroutine which have applications in rounding weighted fractional matchings.