A faster algorithm for packing branchings in digraphs

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We consider the problem of finding an integral (and fractional) packing of branchings in a capacitated digraph with root-set demands. Schrijver described an algorithm that returns a packing with at most m + n(3) + r branchings that makes at most m(m + n3 + r) calls to an oracle that basically computes a minimum cut, where n is the number of vertices, m is the number of arcs and r is the number of root-sets of the input digraph. Leston-Rey and Wakabayashi described an algorithm that returns a packing with at most m + r - 1 branchings but makes a large number of oracle calls. In this work we provide an algorithm, inspired on ideas of Schrijver and in a paper of Gabow and Manu, that returns a packing with at most m+r 1 branchings and makes at most (m+r+2)n oracle calls. Moreover, for the arborescence packing problem our algorithm provides a packing with at most m n + 2 arborescences - thus improving the bound of m of Leston-Rey and Wakabayashi - and makes at most (m - n+5)n oracle calls. (C) 2015 Elsevier B.V. All rights reserved.We consider the problem of finding an integral (and fractional) packing of branchings in a capacitated digraph with root-set demands. Schrijver described an algorithm that returns a packing with at most m + n(3) + r branchings that makes at most m(m + n3194121131CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq [301310/2005-0]CNPq [473867/2010-9, 477692/2012-5]301310/2005-0; 473867/2010-9; 477692/2012-5Wewould like to thank the reviewers for the careful reading of a previous version of this paper and for several suggestions that improved the presentation of the final version. The first author’s research was supported by Bolsa de Produtividade do CNPq P

Some properties of synchrotron radio and inverse-Compton gamma-ray images of supernova remnants

The synchrotron radio maps of supernova remnants (SNRs) in uniform interstellar medium and interstellar magnetic field (ISMF) are analyzed, allowing different `sensitivity' of injection efficiency to the shock obliquity. The very-high energy gamma-ray maps due to inverse Compton process are also synthesized. The properties of images in these different wavelength bands are compared, with particular emphasis on the location of the bright limbs in bilateral SNRs. Recent H.E.S.S. observations of SN 1006 show that the radio and IC gamma-ray limbs coincide, and we found that this may happen if: i) injection is isotropic but the variation of the maximum energy of electrons is rather quick to compensate for differences in magnetic field; ii) obliquity dependence of injection (either quasi-parallel or quasi-perpendicular) and the electron maximum energy is strong enough to dominate magnetic field variation. In the latter case, the obliquity dependence of the injection and the maximum energy should not be opposite. We argue that the position of the limbs alone and even their coincidence in radio, X-rays and gamma-rays, as it is discovered by H.E.S.S. in SN 1006, cannot be conclusive about the dependence of the electron injection efficiency, the compression/amplification of ISMF and the electron maximum energy on the obliquity angle.Comment: Accepted for publication in MNRA

33-anti-circulant digraphs are α\alpha-diperfect and BE-diperfect

Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is exactly on a path of $\mathcal{P}$. We say that a stable set $S$ and a path partition $\mathcal{P}$ are orthogonal if each path of $P$ contains exactly one vertex of $S$. A digraph $D$ satisfies the $\alpha$-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that $S$ and $\mathcal{P}$ are orthogonal. A digraph $D$ is $\alpha$-diperfect if every induced subdigraph of $D$ satisfies the $\alpha$-property. In 1982, Claude Berge proposed a characterization for $\alpha$-diperfect digraphs in terms of forbidden anti-directed odd cycles. In 2018, Sambinelli, Silva and Lee proposed a similar conjecture. A digraph $D$ satisfies the Begin-End-property or BE-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that (i) $S$ and $\mathcal{P}$ are orthogonal and (ii) for each path $P \in \mathcal{P}$, either the start or the end of $P$ belongs to $S$. A digraph $D$ is BE-diperfect if every induced subdigraph of $D$ satisfies the BE-property. Sambinelli, Silva and Lee proposed a characterization for BE-diperfect digraphs in terms of forbidden blocking odd cycles. In this paper, we verified both conjectures for $3$-anti-circulant digraphs. We also present some structural results for $\alpha$-diperfect and BE-diperfect digraphs.Comment: arXiv admin note: substantial text overlap with arXiv:2111.1216

Mach-Zehnder Interferometry in a Strongly Driven Superconducting Qubit

We demonstrate Mach-Zehnder-type interferometry in a superconducting flux qubit. The qubit is a tunable artificial atom, whose ground and excited states exhibit an avoided crossing. Strongly driving the qubit with harmonic excitation sweeps it through the avoided crossing two times per period. As the induced Landau-Zener transitions act as coherent beamsplitters, the accumulated phase between transitions, which varies with microwave amplitude, results in quantum interference fringes for n=1...20 photon transitions. The generalization of optical Mach-Zehnder interferometry, performed in qubit phase space, provides an alternative means to manipulate and characterize the qubit in the strongly-driven regime.Comment: 14 pages, 6 figure