Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.20351

Blossom VI: A Practical Minimum Weight Perfect Matching Algorithm

Pavel Arkhipov·Vladimir Kolmogorov
View original source ↗Read source document ↗Export metadata ↓

Abstract

Minimum weight perfect matching is a fundamental problem in combinatorial optimization. Since 2009, Blossom V has been the leading practical implementation. We present Blossom VI, a practical algorithm that improves upon Blossom V. We test the performance of our implementation on nine benchmark families with graphs containing up to roughly 10 million edges. Blossom VI is significantly faster on the hardest tested instances, while being at most a factor of two slower on families where Blossom V already scales nearly linearly. Blossom VI is a primal-dual solver whose primal phase computes a maximum-cardinality matching in the zero-slack subgraph using cherry trees. At the end of each primal phase, Blossom VI contracts entire cherry blossoms instead of sequences of nested traditional blossoms that would be created by Blossom V. This produces shallower contraction hierarchies and avoids costly cascades of blossom expansions. Our measurements show that reduced blossom depth and fewer expensive expansion operations can explain the observed improvement.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pavel Arkhipov, Vladimir Kolmogorov. 2026-04-22. Blossom VI: A Practical Minimum Weight Perfect Matching Algorithm. https://arxiv.org/abs/2604.20351

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗