Improved Bounds for Fully Dynamic Matching via Ordered Ruzsa-Szemeredi Graphs

TL;DR

This work further strengthens the result of Behnezhad and Ghafari and pushes it to limit to obtain a randomized algorithm with amortized update time of n^{o(1) + O(\epsilon) \cdot ORS(n) with high probability, even against an adaptive adversary.

Abstract

In a very recent breakthrough, Behnezhad and Ghafari [FOCS'24] developed a novel fully dynamic randomized algorithm for maintaining a $(1-ε)$-approximation of maximum matching with amortized update time potentially much better than the trivial $O(n)$ update time. The runtime of the BG algorithm is parameterized via the following graph theoretical concept: * For any $n$, define $ORS(n)$ -- standing for Ordered RS Graph -- to be the largest number of edge-disjoint matchings $M_1,\ldots,M_t$ of size $Θ(n)$ in an $n$-vertex graph such that for every $i \in [t]$, $M_i$ is an induced matching in the subgraph $M_{i} \cup M_{i+1} \cup \ldots \cup M_t$. Then, for any fixed $ε> 0$, the BG algorithm runs in \[ O\left( \sqrt{n^{1+O(ε)} \cdot ORS(n)} \right) \] amortized update time with high probability, even against an adaptive adversary. $ORS(n)$ is a close variant of a more well-known quantity regarding RS graphs (which require every matching to be induced regardless of the ordering). It is currently only known that $n^{o(1)} \leqslant ORS(n) \leqslant n^{1-o(1)}$, and closing this gap appears to be a notoriously challenging problem. In this work, we further strengthen the result of Behnezhad and Ghafari and push it to limit to obtain a randomized algorithm with amortized update time of \[ n^{o(1)} \cdot ORS(n) \] with high probability, even against an adaptive adversary. In the limit, i.e., if current lower bounds for $ORS(n) = n^{o(1)}$ are almost optimal, our algorithm achieves an $n^{o(1)}$ update time for $(1-ε)$-approximation of maximum matching, almost fully resolving this fundamental question. In its current stage also, this fully reduces the algorithmic problem of designing dynamic matching algorithms to a purely combinatorial problem of upper bounding $ORS(n)$ with no algorithmic considerations.

Figures (2)

• Figure 1: An illustration of the three graphs $G_{\textnormal{old}},G_{\textnormal{batch}},G_{\textnormal{match}}$ in \ref{['alg:base-case']}, their role, and how they are being processed. Notice that $G_{\textnormal{old}}$ is a decremental graph, while $G_{\textnormal{batch}},G_{\textnormal{match}}$ are fully dynamic. The analysis of the algorithm forms an ORS from the edges of the matchings $M_1,M_2,\ldots,$ moved from $G_{\textnormal{old}}$ to $G_{\textnormal{match}}$ -- this ORS is a subgraph of the static graph $G$ at the beginning of the batch and does not contain any edges inserted in this batch.
• Figure 2: An illustration of the three graphs $G_{\textnormal{old}},G_{\textnormal{batch}},G_{\textnormal{match}}$ in \ref{['alg:recursive']} for $\textnormal{\large $\mathbb{A}$}_{k+1}$, their role, and how they are being processed. Notice that $G_{\textnormal{old}}$ is a decremental graph, while $G_{\textnormal{batch}},G_{\textnormal{match}}$ are fully dynamic. The main difference with \ref{['alg:base-case']} is that $G_{\textnormal{batch}}$ and $G_{\textnormal{match}}$ are now being handled recursively with $\textnormal{\large $\mathbb{A}$}_k$ (steps $(a)$ and $(b)$ also now involve running \ref{['prop:subtime-size']} to check if applying $\textnormal{\large $\mathbb{A}$}_k$ is valid). This algorithm also form an ORS from the edges of the matchings $M_1,M_2,\ldots,$ moved from $G_{\textnormal{old}}$ to $G_{\textnormal{match}}$.

Theorems & Definitions (33)

• Definition 1.1: Ordered Ruzsa-Szemerédi (ORS) Graphs BehnezhadG24
• Proposition 2.2: McGregor05AhnG11Tirodkar18AssadiLT21BhattacharyaKS23
• Proposition 2.3: Behnezhad21
• Proposition 2.4: AssadiKLY16AssadiKL16ChitnisCEHMMV16Kiss22
• Lemma 2.5
• proof
• Lemma 3.1
• Claim 3.2
• proof
• Claim 3.3