On the Hardness of Computing an Average Curve

We study the complexity of clustering curves under $k$-median and $k$-center objectives in the metric space of the Fr\'echet distance and related distance measures. Building upon recent hardness results for the minimum-enclosing-ball problem under the Fr\'echet distance, we show that also the $1$-median problem is NP-hard. Furthermore, we show that the $1$-median problem is W[1]-hard with the number of curves as parameter. We show this under the discrete and continuous Fr\'echet and Dynamic Time Warping (DTW) distance. This yields an independent proof of an earlier result by Bulteau et al. from 2018 for a variant of DTW that uses squared distances, where the new proof is both simpler and more general. On the positive side, we give approximation algorithms for problem variants where the center curve may have complexity at most $\ell$ under the discrete Fr\'echet distance. In particular, for fixed $k,\ell$ and $\varepsilon$, we give $(1+\varepsilon)$-approximation algorithms for the $(k,\ell)$-median and $(k,\ell)$-center objectives and a polynomial-time exact algorithm for the $(k,\ell)$-center objective

Approximating the Permanent of a Random Matrix with Vanishing Mean

We show an algorithm for computing the permanent of a random matrix with vanishing mean in quasi-polynomial time. Among special cases are the Gaussian, and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time 2^{O(n^{\eps})} for any small constant \eps>0. Our algorithm counters the intuition that the permanent is hard because of the "sign problem" - namely the interference between entries of a matrix with different signs. A major open question then remains whether one can provide an efficient algorithm for random matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the baseline assumptions of the BosonSampling paradigm

Pseudorandomness for Approximate Counting and Sampling

We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions. Our main technical contribution allows one to “boost” a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize Arthur-Merlin games (i.e., show AM = NP) are in fact all equivalent. We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the “boosting” theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM. We observe that Cai's proof that S_2^P ⊆ PP⊆(NP) and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice

XMM-Newton monitoring of X-ray variability in the quasar PKS 0558-504

We present the temporal analysis of X-ray observations of the radio-loud Narrow-Line Seyfert 1 galaxy (NLS1) PKS 0558-504 obtained during the XMM-Newton Calibration and Performance Verification (Cal/PV) phase. The long term light curve is characterized by persistent variability with a clear tendency for the X-ray continuum to harden when the count rate increases. Another strong correlation on long time scales has been found between the variability in the hard band and the total flux. On shorter time scales the most relevant result is the presence of smooth modulations, with characteristic time of ~ 2 hours observed in each individual observation. The short term spectral variability turns out to be rather complex but can be described by a well defined pattern in the hardness ratio-count rate plane.Comment: 6 pages, 7 figures, accepted for publication in A&A special issue on first results from XM

Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

The study of graph products is a major research topic and typically concerns the term $f(G*H)$, e.g., to show that $f(G*H)=f(G)f(H)$. In this paper, we study graph products in a non-standard form $f(R[G*H]$ where $R$ is a "reduction", a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. (1) A tight $n^{1-\epsilon}$-approximation hardness for the minimum consistent deterministic finite automaton (DFA) problem, where $n$ is the sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming $NP\neq RP$ (the weakest possible assumption). (2) A tight $n^{1/2-\epsilon}$ hardness for the edge-disjoint paths (EDP) problem on directed acyclic graphs (DAGs), where $n$ denotes the number of vertices. (3) A tight hardness of packing vertex-disjoint $k$-cycles for large $k$. (4) An alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst.Sci. 2008]

Redox reactions with empirical potentials: Atomistic battery discharge simulations

Batteries are pivotal components in overcoming some of today's greatest technological challenges. Yet to date there is no self-consistent atomistic description of a complete battery. We take first steps toward modeling of a battery as a whole microscopically. Our focus lies on phenomena occurring at the electrode-electrolyte interface which are not easily studied with other methods. We use the redox split-charge equilibration (redoxSQE) method that assigns a discrete ionization state to each atom. Along with exchanging partial charges across bonds, atoms can swap integer charges. With redoxSQE we study the discharge behavior of a nano-battery, and demonstrate that this reproduces the generic properties of a macroscopic battery qualitatively. Examples are the dependence of the battery's capacity on temperature and discharge rate, as well as performance degradation upon recharge.Comment: 14 pages, 10 figure