Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

A skew-symmetric graph $(D=(V,A),\sigma)$ is a directed graph $D$ with an involution $\sigma$ on the set of vertices and arcs. In this paper, we introduce a separation problem, $d$-Skew-Symmetric Multicut, where we are given a skew-symmetric graph $D$, a family of $\cal T$ of $d$-sized subsets of vertices and an integer $k$. The objective is to decide if there is a set $X\subseteq A$ of $k$ arcs such that every set $J$ in the family has a vertex $v$ such that $v$ and $\sigma(v)$ are in different connected components of $D'=(V,A\setminus (X\cup \sigma(X))$. In this paper, we give an algorithm for this problem which runs in time $O((4d)^{k}(m+n+\ell))$, where $m$ is the number of arcs in the graph, $n$ the number of vertices and $\ell$ the length of the family given in the input. Using our algorithm, we show that Almost 2-SAT has an algorithm with running time $O(4^kk^4\ell)$ and we obtain algorithms for {\sc Odd Cycle Transversal} and {\sc Edge Bipartization} which run in time $O(4^kk^4(m+n))$ and $O(4^kk^5(m+n))$ respectively. This resolves an open problem posed by Reed, Smith and Vetta [Operations Research Letters, 2003] and improves upon the earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010]. We also show that Deletion q-Horn Backdoor Set Detection is a special case of 3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor Set Detection which runs in time $O(12^kk^5\ell)$. This gives the first fixed-parameter tractable algorithm for this problem answering a question posed in a paper by a superset of the authors [STACS, 2013]. Using this result, we get an algorithm for Satisfiability which runs in time $O(12^kk^5\ell)$ where $k$ is the size of the smallest q-Horn deletion backdoor set, with $\ell$ being the length of the input formula

Sharp weighted estimates for multi-frequency Calder\'on-Zygmund operators

In this paper we study weighted estimates for the multi-frequency $\omega-$Calder\'{o}n-Zygmund operators $T$ associated with the frequency set $\Theta=\{\xi_1,\xi_2,\dots,\xi_N\}$ and modulus of continuity $\omega$ satisfying the usual Dini condition. We use the modern method of domination by sparse operators and obtain bounds $\|T\|_{L^p(w)\rightarrow L^p(w)}\lesssim N^{|\frac{1}{r}-\frac{1}{2}|}[w]_{\mathbb{A}_{p/r}}^{max(1,\frac{1}{p-r})},~1\leq r<p<\infty,$ for the exponents of $N$ and $\mathbb{A}_{p/r}$ characteristic $[w]_{\mathbb{A}_{p/r}}$

Radiatively Generated νe\nu_e Oscillations: General Analysis, Textures and Models

We study the consequences of assuming that the mass scale $\Delta_{odot}$ corresponding to the solar neutrino oscillations and mixing angle $U_{e3}$ corresponding to the electron neutrino oscillation at CHOOZ are radiatively generated through the standard electroweak gauge interactions. All the leptonic mass matrices having zero $\Delta_{odot}$ and $U_{e3}$ at a high scale lead to a unique low energy value for the $\Delta_{odot}$ which is determined by the (known) size of the radiative corrections, solar and the atmospheric mixing angle and the Majorana mass of the neutrino observed in neutrinoless double beta decay. This prediction leads to the following consequences: ($i$) The MSSM radiative corrections generate only the dark side of the solar neutrino solutions. ($ii$) The inverted mass hierarchy ($m,-m,0$) at the high scale fails in generating the LMA solution but it can lead to the LOW or vacuum solutions. ($iii$) The $\Delta_{odot}$ generated in models with maximal solar mixing at a high scale is zero to the lowest order in the radiative parameter. It tends to get suppressed as a result of this and lies in the vacuum region. We discuss specific textures which can lead to the LMA solution in the present framework and provide a gauge theoretical realization of this in the context of the seesaw model.Comment: 19 pages, LATE

Fast Neutrino Decay in the Minimal Seesaw Model

Neutrino decay in the minimal seesaw model containing three right handed neutrinos and a complex $SU(2)\times U(1)$ singlet Higgs in addition to the standard model fields is considered. A global horizontal symmetry $U(1)_H$ is imposed, which on spontaneous breaking gives rise to a Goldstone boson. This symmetry is chosen in a way that makes a) the contribution of heavy ($\leq$ MeV) majorana neutrinos to the neutrinoless double beta decay amplitude vanish and b) allows the heavy neutrino to decay to a lighter neutrino and the Goldstone boson. It is shown that this decay can occur at a rate much faster than in the original Majoron model even if one does not introduce any additional Higgs fields as is done in the literature. Possibility of describing the 17 keV neutrino in this minimal seesaw model is investigated. While most of the cosmological and astrophysical constraints on the 17 keV neutrino can be satisfied in this model, the laboratory limits coming from the neutrino oscillations cannot be easily met. An extension which removes this inadequacy and offers a consistent description of the 17 keV neutrino is discussed.Comment: 16 pages, PRL-TH/92-1

A Linear Time Parameterized Algorithm for Node Unique Label Cover

The optimization version of the Unique Label Cover problem is at the heart of the Unique Games Conjecture which has played an important role in the proof of several tight inapproximability results. In recent years, this problem has been also studied extensively from the point of view of parameterized complexity. Cygan et al. [FOCS 2012] proved that this problem is fixed-parameter tractable (FPT) and Wahlstr\"om [SODA 2014] gave an FPT algorithm with an improved parameter dependence. Subsequently, Iwata, Wahlstr\"om and Yoshida [2014] proved that the edge version of Unique Label Cover can be solved in linear FPT-time. That is, there is an FPT algorithm whose dependence on the input-size is linear. However, such an algorithm for the node version of the problem was left as an open problem. In this paper, we resolve this question by presenting the first linear-time FPT algorithm for Node Unique Label Cover