Near-Optimal Dynamic Rounding of Fractional Matchings in Bipartite Graphs

TL;DR

The paper advances dynamic rounding of fractional matchings by introducing a simple deterministic dynamic rounding for bipartite graphs with update time $\tilde{O}(\varepsilon^{-1}\log^2 n)$ and a high-probability randomized variant with $\tilde{O}(\varepsilon^{-1}(\log\log n)^2)$ updates, along with an output-adaptive $\tilde{O}(\varepsilon^{-1})$-time option. It then develops a framework of partial rounding and coarsening to speed up dynamic rounding, extending the results to general graphs and to maintaining almost-maximal or AMM structures, with applications to decremental and robust dynamic matching problems. The main approach combines bitwise (binary) rounding with degree-splitting, buffered updates, and coarsening, enabling reductions of rounding to tractable subproblems and yielding near-optimal recourse bounds. These techniques lead to state-of-the-art dynamic matching guarantees across bipartite and general graphs, including deterministic, adaptive, and output-adaptive variants, and they provide a principled route to sub-polynomial update times for decremental settings. Overall, the work significantly tightens the gap between dynamic rounding inefficiency and a known recourse lower bound, while offering practical sublogarithmic and polylogarithmic improvements for a broad class of dynamic matching problems.

Abstract

We study dynamic $(1-ε)$-approximate rounding of fractional matchings -- a key ingredient in numerous breakthroughs in the dynamic graph algorithms literature. Our first contribution is a surprisingly simple deterministic rounding algorithm in bipartite graphs with amortized update time $O(ε^{-1} \log^2 (ε^{-1} \cdot n))$, matching an (unconditional) recourse lower bound of $Ω(ε^{-1})$ up to logarithmic factors. Moreover, this algorithm's update time improves provided the minimum (non-zero) weight in the fractional matching is lower bounded throughout. Combining this algorithm with novel dynamic \emph{partial rounding} algorithms to increase this minimum weight, we obtain several algorithms that improve this dependence on $n$. For example, we give a high-probability randomized algorithm with $\tilde{O}(ε^{-1}\cdot (\log\log n)^2)$-update time against adaptive adversaries. (We use Soft-Oh notation, $\tilde{O}$, to suppress polylogarithmic factors in the argument, i.e., $\tilde{O}(f)=O(f\cdot \mathrm{poly}(\log f))$.) Using our rounding algorithms, we also round known $(1-ε)$-decremental fractional bipartite matching algorithms with no asymptotic overhead, thus improving on state-of-the-art algorithms for the decremental bipartite matching problem. Further, we provide extensions of our results to general graphs and to maintaining almost-maximal matchings.

Theorems & Definitions (104)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4: Informal version of \ref{['thm:general-formal']}
• Theorem 1.5
• proof
• proof
• Proposition 2.3
• Theorem 3.1
• Definition 3.2