(1,j)(1,j)-set problem in graphs

A subset $D \subseteq V$of a graph $G = (V, E)$ is a $(1, j)$-set if every vertex $v \in V \setminus D$ is adjacent to at least $1$ but not more than $j$ vertices in D. The cardinality of a minimum $(1, j)$-set of $G$, denoted as $\gamma_{(1,j)} (G)$, is called the $(1, j)$-domination number of $G$. Given a graph $G = (V, E)$ and an integer $k$, the decision version of the $(1, j)$-set problem is to decide whether $G$ has a $(1, j)$-set of cardinality at most $k$. In this paper, we first obtain an upper bound on $\gamma_{(1,j)} (G)$ using probabilistic methods, for bounded minimum and maximum degree graphs. Our bound is constructive, by the randomized algorithm of Moser and Tardos [MT10], We also show that the $(1, j)$- set problem is NP-complete for chordal graphs. Finally, we design two algorithms for finding $\gamma_{(1,j)} (G)$ of a tree and a split graph, for any fixed $j$, which answers an open question posed in [CHHM13]

Posterior Contraction rate for one group global-local shrinkage priors in sparse normal means problem

We consider a high-dimensional sparse normal means model where the goal is to estimate the mean vector assuming the proportion of non-zero means is unknown. Using a Bayesian setting, we model the mean vector by a one-group global-local shrinkage prior belonging to a broad class of such priors that includes the horseshoe prior. We address some questions related to asymptotic properties of the resulting posterior distribution of the mean vector for the said class priors. Since the global shrinkage parameter plays a pivotal role in capturing the sparsity in the model, we consider two ways to model this parameter in this paper. Firstly, we consider this as an unknown fixed parameter and estimate it by an empirical Bayes estimate. In the second approach, we do a hierarchical Bayes treatment by assigning a suitable non-degenerate prior distribution to it. We first show that for the class of priors under study, the posterior distribution of the mean vector contracts around the true parameter at a near minimax rate when the empirical Bayes approach is used. Next, we prove that in the hierarchical Bayes approach, the corresponding Bayes estimate attains the minimax risk asymptotically under the squared error loss function. We also show that the posterior contracts around the true parameter at a near minimax rate. These results generalize those of van der Pas et al. (2014) \cite{van2014horseshoe}, (2017) \cite{van2017adaptive}, proved for the horseshoe prior. We have also studied in this work the asymptotic optimality of the horseshoe+ prior to this context. For horseshoe+ prior, we prove that using the empirical Bayes estimate of the global parameter, the corresponding Bayes estimate attains the near minimax risk asymptotically under the squared error loss function and also shows that the posterior distribution contracts around the true parameter at a near minimax rate

Uniformity of point samples in metric spaces using gap ratio

Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We also show some interesting connections of gap ratio to Delaunay triangulation and discrepancy in the Euclidean plane. The major focus of this work is on solving optimization related questions about selecting uniform point samples from metric spaces; the uniformity being measured using gap ratio. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. We deduce lower bounds, prove hardness and approximation hardness results. We show that a general approximation algorithm framework gives different approximation ratios for different metric spaces based on the lower bound we deduce. Apart from the above, we show existence of coresets for sampling uniform points from the Euclidean space -- for both the static and the streaming case. This leads to a $\left( 1+\epsilon \right)$-approximation algorithm for uniform sampling from the Euclidean space.Comment: 31 pages, 10 figure

A multilevel approach for assessing business strategies on climate change

The need for an interdisciplinary and integrative approach for doing research on business strategies and climate change is gaining increasing recognition. However, there is a consensus that such crossfertilization is currently missing. Multilevel research methods by virtue of being interdisciplinary in nature may address this need. This paper proposes to advance the adoption of multilevel research approach in the context of business strategies and climate change. As a guide for conducting multilevel assessment, a flexible analytical framework is presented. The framework is developed through a process of structured literature review. The framework consists of thirteen contextual factors spread across five levels and identifies the key multilevel relationships that moderate organisational level climate change related strategy formulation. Level specificities of several theories across these five levels are also identified to facilitate application of the framework in building multilevel hypotheses for business strategies on climate change. In addition, a concise summary of the fundamental concepts of multilevel modelling techniques is provided to help researchers in selecting suitable multilevel models during the operationalization of the framework. The operationalization of the framework is demonstrated by building and testing a three level hypotheses on corporate lobbying activities on climate change issues. It is observed that irrespective of their locations, financially underperforming companies with a larger workforce and belonging to sectors with higher Green House Gas emission intensities particularly lobby intensely on climate change issues. In conclusion, the potential challenges and opportunities in applying the framework for building multilevel theories in the context of business strategies and climate change are discussed. (C) 2017 The Authors. Published by Elsevier Ltd

An optimal and a heuristic approach to solve the route and spectrum allocation problem in OFDM networks

To maximize the usage of optical resources, it is important to reduce the total bandwidth requirement for communication. Orthogonal Frequency Division Multiplexing (OFDM) has recently emerged as an encouraging competitor to Wavelength Division Multiplexing (WDM), which uses fixed capacity channels. A network using OFDM-based Spectrum-sliced Elastic Optical Path (SLICE) has a higher spectrum efficiency, due to the fine granularity of subcarrier frequencies used. To minimize the utilized spectrum in SLICE networks, the routing and spectrum allocation problem (RSA) has to be efficiently solved. We have solved the RSA problem using two Integer Linear Programming (ILP) formulations. Our first formulation provides an optimal solution, based on an exhaustive search and is useful as a benchmark. Our second approach reduces the time requirement by restricting the number of paths considered for each commodity, without significantly compromising on the solution quality. We have compared our approaches with another prominent formulation proposed recently

Effects due to unconventional pairing in transport through a normal metalsuperconductor-normal metal hybrid junction

We explore transport properties of a normal metal-superconductor-normal metal (NSN) junction, where the superconducting region supports mixed singlet and chiral triplet pairings. We show that in the subgapped regime when the chiral triplet pairing amplitude dominates over that of the singlet, a resonance phenomena emerges out where all the quantum mechanical scattering probabilities acquire a value of 0.25. At the resonance, crossed Andreev reflection mediating through such junction, acquires a zero energy peak. This reflects as a zero energy peak in the conductance as well in the topological phase when $\Delta_p > \Delta_s$.Comment: Conference Proceeding