FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.Comment: 16 pages, 4 figures; to appear in Mathematical Programmin

The Complexity of Drawing Graphs on Few Lines and Few Planes

It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the minimum number of lines in $\mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. - For $d\in\{2,3\}$, we prove that deciding whether $\rho^1_d(G)\le k$ for a given graph $G$ and integer $k$ is ${\exists\mathbb{R}}$-complete. - Since $\mathrm{NP}\subseteq{\exists\mathbb{R}}$, deciding $\rho^1_d(G)\le k$ is NP-hard for $d\in\{2,3\}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. - Since ${\exists\mathbb{R}}\subseteq\mathrm{PSPACE}$, both $\rho^1_2(G)$ and $\rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $\rho^1_2$ or $\rho^1_3$ sometimes require irrational coordinates. - Let $\rho^2_3(G)$ be the minimum number of planes in $\mathbb{R}^3$ needed to cover a straight-line drawing of a graph $G$. We prove that deciding whether $\rho^2_3(G)\le k$ is NP-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $\mathrm{P}=\mathrm{NP}$

On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I

PLEASE NOTE: The original Technical Report TR00853 is missing. A copy can be found at http://www.sciencedirect.com/science/article/pii/S0747717110800033This paper published in the "Journal of Symbolic Computation" 13 (1992) 255-35

Parts of Quantum States

It is shown that generic N-party pure quantum states (with equidimensional subsystems) are uniquely determined by their reduced states of just over half the parties; in other words, all the information in almost all N-party pure states is in the set of reduced states of just over half the parties. For N even, the reduced states in fewer than N/2 parties are shown to be an insufficient description of almost all states (similar results hold when N is odd). It is noted that Real Algebraic Geometry is a natural framework for any analysis of parts of quantum states: two simple polynomials, a quadratic and a cubic, contain all of their structure. Algorithmic techniques are described which can provide conditions for sets of reduced states to belong to pure or mixed states.Comment: 10 pages, 1 figur

Ultrastructural and Histological Analysis of Dark Spot Syndrome in Siderastrea siderea and Agaricia agaricites

Dark Spot Syndrome (DSS) typically manifests in scleractinian corals as lesions of varying color, size, shape and location that can result in skeletal changes and tissue death. A causative agent for DSS has not yet been identified. The objective of this study was histological and ultrastructural comparison of the cellular and skeletal characteristics of DSS-affected and healthy Siderastrea siderea and Agaricia agaricites. The greater resolution possible with transmission electron microscopy (TEM) revealed microbial activity and tissue changes not resolvable utilizing histology. DSS-affected tissue had less integrity, with increasing cellular degradation and vacuolization. A high concentration of electron dense inclusions, which appear to be zymogen granules, was concentrated in the calicodermis and adjacent gastrodermal layer. Numerous endolithic fungal cells were observed directly adjacent to the calicodermis in DSS-affected S. siderea. Numerous unidentified endolithic cells were observed directly adjacent to the calicodermis in DSS-affected A. agaricites. These observations suggest that the coral may be using a digestive enzyme as a defensive mechanism against endolithic cellular invasion

Semidefinite Representation of the kk-Ellipse

The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$ if $k$ is odd and degree $2^k{-}\binom{k}{k/2}$ if $k$ is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted $k$-ellipses and $k$-ellipsoids in arbitrary dimensions, and it leads to new geometric applications of semidefinite programming.Comment: 16 pages, 5 figure

Realizability of Polytopes as a Low Rank Matrix Completion Problem

This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another set parameterizing the projective moduli space of a combinatorial polytope

The Complexity of Nash Equilibria in Simple Stochastic Multiplayer Games

We analyse the computational complexity of finding Nash equilibria in simple stochastic multiplayer games. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game G, does there exist a pure-strategy Nash equilibrium of G where player 0 wins with probability 1. Moreover, this problem remains undecidable if it is restricted to strategies with (unbounded) finite memory. However, if mixed strategies are allowed, decidability remains an open problem. One way to obtain a provably decidable variant of the problem is restricting the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively.Comment: 23 pages; revised versio

Stability and Complexity of Minimising Probabilistic Automata

We consider the state-minimisation problem for weighted and probabilistic automata. We provide a numerically stable polynomial-time minimisation algorithm for weighted automata, with guaranteed bounds on the numerical error when run with floating-point arithmetic. Our algorithm can also be used for "lossy" minimisation with bounded error. We show an application in image compression. In the second part of the paper we study the complexity of the minimisation problem for probabilistic automata. We prove that the problem is NP-hard and in PSPACE, improving a recent EXPTIME-result.Comment: This is the full version of an ICALP'14 pape

Cryotomography of budding influenza a virus reveals filaments with diverse morphologies that mostly do not bear a genome at their distal end

Influenza viruses exhibit striking variations in particle morphology between strains. Clinical isolates of influenza A virus have been shown to produce long filamentous particles while laboratory-adapted strains are predominantly spherical. However, the role of the filamentous phenotype in the influenza virus infectious cycle remains undetermined. We used cryo-electron tomography to conduct the first three-dimensional study of filamentous virus ultrastructure in particles budding from infected cells. Filaments were often longer than 10 microns and sometimes had bulbous heads at their leading ends, some of which contained tubules we attribute to M1 while none had recognisable ribonucleoprotein (RNP) and hence genome segments. Long filaments that did not have bulbs were infrequently seen to bear an ordered complement of RNPs at their distal ends. Imaging of purified virus also revealed diverse filament morphologies; short rods (bacilliform virions) and longer filaments. Bacilliform virions contained an ordered complement of RNPs while longer filamentous particles were narrower and mostly appeared to lack this feature, but often contained fibrillar material along their entire length. The important ultrastructural differences between these diverse classes of particles raise the possibility of distinct morphogenetic pathways and functions during the infectious process