Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings

Marek Cygan; Harold N. Gabow; Piotr Sankowski

Home/Archive/arXiv

arXiv · 1204.1616

Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings

Marek Cygan·Harold N. Gabow·Piotr Sankowski

View original source ↗Read source document ↗Export metadata ↓

Abstract

Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen's theorem and of Strojohann's determinant algorithm. It allows us to give new and simple solutions to the following problems: * Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^{\omega}) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with non-negative weights) this matches the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the other hand, no algorithm working in \tilde{O}(Wn^{\omega}) time was previously known for undirected graphs with negative weights. Furthermore our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. * Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^{\omega}) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge no algorithm with this running time was known for undirected graphs with negative weights. * Finding Minimum Weight Perfect Matchings -- We present an \tilde{O}(Wn^{\omega}) time algorithm for finding minimum weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski in 2006, who presented such an algorithm but only in the case of bipartite graphs. In order to solve minimum weight perfect matching problem we develop a novel combinatorial interpretation of the dual solution which sheds new light on this problem. Such a combinatorial interpretation was not know previously, and is of independent interest.

Explore related subjects

cs.DS

Keep this discovery

BibTeX RIS

Marek Cygan, Harold N. Gabow, Piotr Sankowski. 2012-04-07. Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings. https://arxiv.org/abs/1204.1616

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

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

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↗