In the graphic matroid parity problem, we are given an undirected multigraph $G=(V,E)$ with $n$ vertices, together with a partition of $E$ into $m$ pairs of distinct edges, and the goal is to find a maximum-cardinality collection of pairs whose union is a forest. This problem generalizes both graph matching and graphic matroid intersection. Graph matching admits the celebrated $\tilde{O}(\sqrt{n}m)$-time algorithm of Micali--Vazirani~[FOCS'80], and graphic matroid intersection admits $o(nm)$-time algorithms due to Gabow--Stallmann~[ICALP'85] and Gabow--Xu~[FOCS'89]. In contrast, for graphic matroid parity, Gabow--Stallmann~[FOCS'84 \& Combinatorica'86] gave an $O(n^2 m)$-time algorithm, and its $\tilde{O}(nm)$-time implementation by Gabow--Stallmann~[ICALP'85] had long remained the best known bound, except in very dense regimes where $m=ω(n^3)$, in which case the algebraic $\tilde{O}(n^4)$-time algorithm of Cheung--Lau--Leung~[SODA'11 \& TALG'14] is faster. We present a randomized $\tilde{O}(n^{3/2} \sqrt{m} + \sqrt{n} m)$-time algorithm for graphic matroid parity. This is the first algorithm to break the $O(nm)$-time complexity barrier for this problem outside the very dense regime. Moreover, it improves over the previous bounds except in very sparse regimes where $m=\tilde{O}(n)$ and in very dense regimes where $m = \tildeΩ(n^{3.5})$. In particular, when $m=Θ(n^2)$, our algorithm runs in $\tilde{O}(n^{2.5})$ time, which nearly matches the $\tilde{O}(n^{2.5})$-time bounds of Micali--Vazirani for graph matching and of Gabow--Xu for graphic matroid intersection in this regime.

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