On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.Comment: 21 pages, Conference version to appear in ESA 201

History of EISCAT – Part 6: the participation of Japan in the EISCAT Scientific Association

In Sect. 1, the original planning of Japanese Svalbard IS (incoherent scatter) radar with phased-array antennas is described. In 1988, this plan was proposed as one of the major projects for the forthcoming Solar–Terrestrial Environment Laboratory, Nagoya University, Japan, to be reorganized by the Research Institute of Atmospherics at Nagoya University. On the other hand, in 1989, UK scientists proposed a plan of polar cap radar with parabolic dish antennas in Longyearbyen to the EISCAT (European incoherent scatter) Council. In Sect. 2, the circumstances leading to Japan's participation in the EISCAT Scientific Association, with details of its processes with strong collaborations with Norwegian scientists and the EISCAT Scientific Association are described. In 1995, Japan participated EISCAT Scientific Association as the seventh member country with funds contributing to the second dish antenna of the EISCAT Svalbard Radar. In Sect. 3, a summary of the EISCAT-related achievement by Japanese scientists is described, where major interests are the lower thermosphere wind dynamics, the magnetosphere–ionosphere–thermosphere coupling, characteristics, and driving mechanisms of ion upflow, electrodynamics of current, electric field and particles, characteristics and production mechanisms of auroras, such as pulsating aurora, and aurora tomography. In Sect. 4, a summary of the scientific collaborations between Japan and Europe, particularly those between Japan and Norway, and hopes for the forthcoming EISCAT_3D and further collaboration with EISCAT community are described.</p

Fast Distributed Approximation for Max-Cut

Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the $\mathcal{LOCAL}$ or $\mathcal{CONGEST}$ models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any $\epsilon > 0$, we develop randomized approximation algorithms achieving a ratio of $(1-\epsilon)$ to the optimum for Max-Cut on bipartite graphs in the $\mathcal{CONGEST}$ model, and on general graphs in the $\mathcal{LOCAL}$ model. We further present efficient deterministic algorithms, including a $1/3$-approximation for Max-Dicut in our models, thus improving the best known (randomized) ratio of $1/4$. Our algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest

ALMA observations of molecules in Supernova 1987A

Supernova (SN) 1987A has provided a unique opportunity to study how SN ejecta evolve in 30 years time scale. We report our ALMA spectral observations of SN 1987A, taken in 2014, 2015 and 2016, with detections of CO, 28SiO, HCO+ and SO, with weaker lines of 29SiO. We find a dip in the SiO line profiles, suggesting that the ejecta morphology is likely elongated. The difference of the CO and SiO line profiles is consistent with hydrodynamic simulations, which show that Rayleigh-Taylor instabilities causes mixing of gas, with heavier elements much more disturbed, making more elongated structure. Using 28SiO and its isotopologues, Si isotope ratios were estimated for the first time in SN 1987A. The estimated ratios appear to be consistent with theoretical predictions of inefficient formation of neutron rich atoms at lower metallicity, such as observed in the Large Magellanic Cloud (about half a solar metallicity). The deduced large HCO+ mass and small SiS mass, which are inconsistent to the predictions of chemical model, might be explained by some mixing of elements immediately after the explosion. The mixing might have made some hydrogen from the envelope to sink into carbon and oxygen-rich zone during early days after the explosion, enabling the formation of a substantial mass of HCO+. Oxygen atoms may penetrate into silicon and sulphur zone, suppressing formation of SiS. Our ALMA observations open up a new window to investigate chemistry, dynamics and explosive-nucleosynthesis in supernovae