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

Deep Chandra Observations of HCG 16 - I. Active Nuclei, Star formation and Galactic Winds

We present new, deep Chandra X-ray and Giant Metrewave Radio Telescope 610~MHz observations of the spiral-galaxy-rich compact group HCG 16, which we use to examine nuclear activity, star formation and the high luminosity X-ray binary populations in the major galaxies. We confirm the presence of obscured active nuclei in NGC 833 and NGC 835, and identify a previously unrecognized nuclear source in NGC 838. All three nuclei are variable on timescales of months to years, and for NGC 833 and NGC 835 this is most likely caused by changes in accretion rate. The deep Chandra observations allow us to detect for the first time an Fe-K$\alpha$ emission line in the spectrum of the Seyfert 2 nucleus of NGC 835. We find that NGC 838 and NGC 839 are both starburst-dominated systems, with only weak nuclear activity, in agreement with previous optical studies. We estimate the star formation rates in the two galaxies from their X-ray and radio emission, and compare these results with estimates from the infra-red and ultra-violet bands to confirm that star formation in both galaxies is probably declining after galaxy-wide starbursts were triggered ~400-500 Myr ago. We examine the physical properties of their galactic superwinds, and find that both have temperatures of ~0.8 keV. We also examine the X-ray and radio properties of NGC 848, the fifth largest galaxy in the group, and show that it is dominated by emission from its starburst.Comment: 18 pages, 11 figures, 11 tables, accepted for publication in ApJ; updated references and fixed typos identified at proof stag

High rate locally-correctable and locally-testable codes with sub-polynomial query complexity

In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length $n$, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity $\exp(\tilde{O}(\sqrt{\log n}))$. Previously such codes were known to exist only with $\Omega(n^{\beta})$ query complexity (for constant $\beta > 0$), and there were several, quite different, constructions known. Our codes are based on a general distance-amplification method of Alon and Luby~\cite{AL96_codes}. We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of \emph{sub-constant relative distance}. Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity $\exp(\tilde{O}(\sqrt{\log n}))$, which additionally have the property of approaching the Singleton bound: they have almost the best-possible relationship between their rate and distance. This has the surprising consequence that asking for a large alphabet error-correcting code to further be an LCC or LTC with $\exp(\tilde{O}(\sqrt{\log n}))$ query complexity does not require any sacrifice in terms of rate and distance! Such a result was previously not known for any $o(n)$ query complexity. Our results on LCCs also immediately give locally-decodable codes (LDCs) with the same parameters

On the NP-Hardness of Approximating Ordering Constraint Satisfaction Problems

We show improved NP-hardness of approximating Ordering Constraint Satisfaction Problems (OCSPs). For the two most well-studied OCSPs, Maximum Acyclic Subgraph and Maximum Betweenness, we prove inapproximability of $14/15+\epsilon$ and $1/2+\epsilon$. An OCSP is said to be approximation resistant if it is hard to approximate better than taking a uniformly random ordering. We prove that the Maximum Non-Betweenness Problem is approximation resistant and that there are width-$m$ approximation-resistant OCSPs accepting only a fraction $1 / (m/2)!$ of assignments. These results provide the first examples of approximation-resistant OCSPs subject only to P $\neq$ \NP

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

Rank deficiency of Kalman error covariance matrices in linear time-varying system with deterministic evolution

We prove that for-linear, discrete, time-varying, deterministic system (perfect-model) with noisy outputs, the Riccati transformation in the Kalman filter asymptotically bounds the rank of the forecast and the analysis error covariance matrices to be less than or equal to the number of nonnegative Lyapunov exponents of the system. Further, the support of these error covariance matrices is shown to be confined to the space spanned by the unstable-neutral backward Lyapunov vectors, providing the theoretical justification for the methodology of the algorithms that perform assimilation only in the unstable-neutral subspace. The equivalent property of the autonomous system is investigated as a special case

A Hypergraph Dictatorship Test with Perfect Completeness

A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based \PCP construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are $o(1).$ Their test makes $q\geq3$ queries and has amortized query complexity $1+O(\frac{\log q}{q})$ but has an inherent loss of perfect completeness. In this paper we give an adaptive hypergraph dictatorship test that achieves both perfect completeness and amortized query complexity $1+O(\frac{\log q}{q})$.Comment: Some minor correction

Elastic electron scattering from 3-hydroxytetrahydrofuran: experimental and theoretical studies

We report the results of measurements and calculations for elastic electron scattering from 3-hydroxytetrahydrofuran (C4H8O2). The measurements are performed with a crossed electron-target beam apparatus and the absolute cross-sections are determined using the relative flow technique. The calculations are carried out using the Schwinger multichannel method in the static-exchange plus polarization (SEP) approximation. A set of angular differential cross-sections (DCS) is provided at five incident energies (6.5, 8, 10, 15 and 20 eV) over an angular range of 20–130°, and the energy dependence of the elastic DCS at a scattering angle of 120° is also presented. Integral elastic and elastic momentum transfer cross-sections have also been derived and calculated. The results are compared with those of recent measurements and calculations for the structurally similar molecule tetrahydrofuran (C4H8O)

Immunoexpression Of Cd95 In Chronic Gastritis And Gastric Mucosa-associated Lymphomas.

CD95 (Fas/APO-1)-mediated apoptosis plays an important role in immunological regulation and is related to the pathogenesis of autoimmune diseases. Immunoexpression of CD95 has been reported to frequently occur in low grade non-Hodgkin lymphomas, especially of post-germinal center histogenesis, among which those originating in mucosa-associated lymphoid tissue (MALT lymphomas). However, there is no report comparing in situ immunoexpression of this marker in lymphomas and the hyperplastic lymphoid reaction (chronic gastritis) related to Helicobacter pylori infection. The purpose of the present research was to compare the intensity of lymphoid CD95 immunoexpression in 15 cases of H. pylori-related chronic gastritis and 15 gastric MALT lymphomas. CD95 (anti-CD95) was detected by an immunoperoxidase technique in paraffin sections using the catalyzed amplification system. Graduation of reaction intensity (percentage of CD95-positive cells) was semiquantitative, from 1+ to 4+. Nine cases of chronic gastritis were 4+, five 2+ and one 1+. Three lymphomas were 4+, three 3+, four 2+, four 1+, and one was negative. Although 14 of 15 lymphomas were positive for CD95, the intensity of the reaction was significantly weaker compared to that obtained with gastric tissue for patients with gastritis (P = 0.03). The difference in CD95 immunoexpression does not seem to be useful as an isolated criterion in the differential diagnosis between chronic gastritis and MALT lymphomas since there was overlapping of immunostaining patterns. However, it suggests the possibility of a pathogenetic role of this apoptosis-regulating protein in MALT lymphomas.371397-40

Free randomness can be amplified

Are there fundamentally random processes in nature? Theoretical predictions, confirmed experimentally, such as the violation of Bell inequalities, point to an affirmative answer. However, these results are based on the assumption that measurement settings can be chosen freely at random, so assume the existence of perfectly free random processes from the outset. Here we consider a scenario in which this assumption is weakened and show that partially free random bits can be amplified to make arbitrarily free ones. More precisely, given a source of random bits whose correlation with other variables is below a certain threshold, we propose a procedure for generating fresh random bits that are virtually uncorrelated with all other variables. We also conjecture that such procedures exist for any non-trivial threshold. Our result is based solely on the no-signalling principle, which is necessary for the existence of free randomness.Comment: 5+7 pages, 2 figures. Updated to match published versio