Hardness Amplification of Optimization Problems

In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products. We say that an optimization problem ? is direct product feasible if it is possible to efficiently aggregate any k instances of ? and form one large instance of ? such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the k smaller instances. Given a direct product feasible optimization problem ?, our hardness amplification theorem may be informally stated as follows: If there is a distribution D over instances of ? of size n such that every randomized algorithm running in time t(n) fails to solve ? on 1/?(n) fraction of inputs sampled from D, then, assuming some relationships on ?(n) and t(n), there is a distribution D\u27 over instances of ? of size O(n??(n)) such that every randomized algorithm running in time t(n)/poly(?(n)) fails to solve ? on 99/100 fraction of inputs sampled from D\u27. As a consequence of the above theorem, we show hardness amplification of problems in various classes such as NP-hard problems like Max-Clique, Knapsack, and Max-SAT, problems in P such as Longest Common Subsequence, Edit Distance, Matrix Multiplication, and even problems in TFNP such as Factoring and computing Nash equilibrium

An Efficient Representation for Filtrations of Simplicial Complexes

A filtration over a simplicial complex $K$ is an ordering of the simplices of $K$ such that all prefixes in the ordering are subcomplexes of $K$. Filtrations are at the core of Persistent Homology, a major tool in Topological Data Analysis. In order to represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree [Algorithmica '14]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest. In this paper, we propose a new data structure called the Critical Simplex Diagram (CSD) which is a variant of the Simplex Array List (SAL) [Algorithmica '17]. Our data structure allows one to store in a compact way the filtration of a simplicial complex, and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Finally, we show that the CSD representation admits fast construction algorithms for Flag complexes and relaxed Delaunay complexes.Comment: A preliminary version appeared in SODA 201

Building Efficient and Compact Data Structures for Simplicial Complexes

The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the Simplex Tree while retaining its functionalities. In addition, we propose two new data structures called the Maximal Simplex Tree (MxST) and the Simplex Array List (SAL). We analyze the compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List under various settings.Comment: An extended abstract appeared in the proceedings of SoCG 201

On the Sensitivity Conjecture for Disjunctive Normal Forms

The sensitivity conjecture of Nisan and Szegedy [CC\u2794] asks whether for any Boolean function f, the maximum sensitivity s(f), is polynomially related to its block sensitivity bs(f), and hence to other major complexity measures. Despite major advances in the analysis of Boolean functions over the last decade, the problem remains widely open. In this paper, we consider a restriction on the class of Boolean functions through a model of computation (DNF), and refer to the functions adhering to this restriction as admitting the Normalized Block property. We prove that for any function f admitting the Normalized Block property, bs(f) <= 4 * s(f)^2. We note that (almost) all the functions mentioned in literature that achieve a quadratic separation between sensitivity and block sensitivity admit the Normalized Block property. Recently, Gopalan et al. [ITCS\u2716] showed that every Boolean function f is uniquely specified by its values on a Hamming ball of radius at most 2 * s(f). We extend this result and also construct examples of Boolean functions which provide the matching lower bounds

Evolution of isolated turbulent trailing vortices

In this work, the temporal evolution of a low swirl-number turbulent Batchelor vortex is studied using pseudospectral direct numerical simulations. The solution of the governing equations in the vorticity-velocity form allows for accurate application of boundary conditions. The physics of the evolution is investigated with an emphasis on the mechanisms that influence the transport of axial and angular momentum. Excitation of normal mode instabilities gives rise to coherent large scale helical structures inside the vortical core. The radial growth of these helical structures and the action of axial shear and differential rotation results in the creation of a polarized vortex layer. This vortex layer evolves into a series of hairpin-shaped structures that subsequently breakdown into elongated fine scale vortices. Ultimately, the radially outward propagation of these structures results in the relaxation of the flow towards a stable high-swirl configuration. Two conserved quantities, based on the deviation from the laminar solution, are derived and these prove to be useful in characterizing the polarized vortex layer and enhancing the understanding of the transport process. The generation and evolution of the Reynolds stresses is also addressed

Systemic inflammatory response syndrome (SIRS) after extracorporeal membrane oxygenation (ECMO): Incidence, risks and survivals.

INTRODUCTION: Systemic inflammatory response syndrome (SIRS) is frequently observed after extracorporeal membrane oxygenation (ECMO) decannulation; however, these issues have not been investigated well in the past. METHODS: Retrospective chart review was performed to identify post-ECMO SIRS phenomenon, defined by exhibiting 2/3 of the following criteria: fever, leukocytosis, and escalation of vasopressors. The patients were divided into 2 groups: patients with documented infections (Group I) and patients with true SIRS (Group TS) without any evidence of infection. Survival and pre-, intra- and post-ECMO risk factors were analyzed. RESULTS: Among 62 ECMO survivors, 37 (60%) patients developed the post-ECMO SIRS phenomenon, including Group I (n = 22) and Group TS (n = 15). The 30-day survival rate of Group I and TS was 77% and 100%, respectively (p = 0.047), although risk factors were identical. CONCLUSIONS: SIRS phenomenon after ECMO decannulation commonly occurs. Differentiating between the similar clinical presentations of SIRS and infection is important and will impact clinical outcomes

Multifarious transparent glass nanocrystal composites

Glasses comprising well known ferroelectric crystalline phases have been a subject of curious investigation from the point of view of exploiting these composites for dielectric, pyroelectric, ferroelectric, electro and non-linear optical devices. Transparent glass-ceramics containing ferroelectric crystallites at nano scale have been of much interest owing to their promising physical properties. The advantages that are associated with glass-ceramics include very low levels of porosity and hence high break down voltages. It is of our interest to nanocrystallize Aurivillius family of ferroelectric oxides and tetragonal tungsten bronzes on borate and tellurite based glass matrices and demonstrate their promising optical and nonlinear optical properties. Apart from the above, the nanocrystallites of well known ferroelectric material LiNbO3 was grown in a reactive glass matrix. These nanocrystals of LiNbO3 exhibited intense second harmonic signals in transmission mode when exposed to IR light at 1064 nm. The most interesting result was the demonstration of optical diffraction of the second harmonic signals which was attributed to the presence of self- organized sub-micrometer sized LiNbO3 crystallites that were indeed inscribed by the IR laser light which was used to probe in the NLO property of these materials