A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits

In this paper we study the complexity of constructing a hitting set for the closure of VP, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given n,s,r in unary outputs a set of n-tuples over the rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all n-variate polynomials of degree-r that are the limit of size-s algebraic circuits. Previously it was known that a random set of this size is a hitting set, but a construction that is certified to work was only known in EXPSPACE (or EXPH assuming the generalized Riemann hypothesis). As a corollary we get that a host of other algebraic problems such as Noether Normalization Lemma, can also be solved in PSPACE deterministically, where earlier only randomized algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann hypothesis) were known. The proof relies on the new notion of a robust hitting set which is a set of inputs such that any nonzero polynomial that can be computed by a polynomial size algebraic circuit, evaluates to a not too small value on at least one element of the set. Proving the existence of such a robust hitting set is the main technical difficulty in the proof. Our proof uses anti-concentration results for polynomials, basic tools from algebraic geometry and the existential theory of the reals

Exponential Time Complexity of Weighted Counting of Independent Sets

We consider weighted counting of independent sets using a rational weight x: Given a graph with n vertices, count its independent sets such that each set of size k contributes x^k. This is equivalent to computation of the partition function of the lattice gas with hard-core self-repulsion and hard-core pair interaction. We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time 2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in time 2^{cn}), counting the independent sets of size n/3 of an n-vertex graph needs time 2^{\Omega(n)} and weighted counting of independent sets needs time 2^{\Omega(n/log^3 n)} for all rational weights x\neq 0. We have two technical ingredients: The first is a reduction from 3SAT to independent sets that preserves the number of solutions and increases the instance size only by a constant factor. Second, we devise a combination of vertex cloning and path addition. This graph transformation allows us to adapt a recent technique by Dell, Husfeldt, and Wahlen which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.Comment: Introduction revised, differences between versions of counting independent sets stated more precisely, minor improvements. 14 page

Experimental discrimination of ion stopping models near the Bragg peak in highly ionized matter

The energy deposition of ions in dense plasmas is a key process in inertial confinement fusion that determines the α-particle heating expected to trigger a burn wave in the hydrogen pellet and resulting in high thermonuclear gain. However, measurements of ion stopping in plasmas are scarce and mostly restricted to high ion velocities where theory agrees with the data. Here, we report experimental data at low projectile velocities near the Bragg peak, where the stopping force reaches its maximum. This parameter range features the largest theoretical uncertainties and conclusive data are missing until today. The precision of our measurements, combined with a reliable knowledge of the plasma parameters, allows to disprove several standard models for the stopping power for beam velocities typically encountered in inertial fusion. On the other hand, our data support theories that include a detailed treatment of strong ion-electron collisions

An Introduction to Temporal Graphs: An Algorithmic Perspective

A \emph{temporal graph} is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence $G_1,G_2\ldots,G_l$ of static graphs over the same (static) set of nodes $V$. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension in added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graph theory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community

Is Information-Theoretic Topology-Hiding Computation Possible?

Topology-hiding computation (THC) is a form of multi-party computation over an incomplete communication graph that maintains the privacy of the underlying graph topology. Existing THC protocols consider an adversary that may corrupt an arbitrary number of parties, and rely on cryptographic assumptions such as DDH. In this paper we address the question of whether information-theoretic THC can be achieved by taking advantage of an honest majority. In contrast to the standard MPC setting, this problem has remained open in the topology-hiding realm, even for simple privacy-free functions like broadcast, and even when considering only semi-honest corruptions. We uncover a rich landscape of both positive and negative answers to the above question, showing that what types of graphs are used and how they are selected is an important factor in determining the feasibility of hiding topology information-theoretically. In particular, our results include the following. We show that topology-hiding broadcast (THB) on a line with four nodes, secure against a single semi-honest corruption, implies key agreement. This result extends to broader classes of graphs, e.g., THB on a cycle with two semi-honest corruptions. On the other hand, we provide the first feasibility result for information-theoretic THC: for the class of cycle graphs, with a single semi-honest corruption. Given the strong impossibilities, we put forth a weaker definition of distributional-THC, where the graph is selected from some distribution (as opposed to worst-case). We present a formal separation between the definitions, by showing a distribution for which information theoretic distributional-THC is possible, but even topology-hiding broadcast is not possible information-theoretically with the standard definition. We demonstrate the power of our new definition via a new connection to adaptively secure low-locality MPC, where distributional-THC enables parties to reuse a secret low-degree communication graph even in the face of adaptive corruptions

A Draft of the Human Septin Interactome

Background: Septins belong to the GTPase superclass of proteins and have been functionally implicated in cytokinesis and the maintenance of cellular morphology. They are found in all eukaryotes, except in plants. In mammals, 14 septins have been described that can be divided into four groups. It has been shown that mammalian septins can engage in homo- and heterooligomeric assemblies, in the form of filaments, which have as a basic unit a hetero-trimeric core. In addition, it has been speculated that the septin filaments may serve as scaffolds for the recruitment of additional proteins. Methodology/Principal Findings: Here, we performed yeast two-hybrid screens with human septins 1-10, which include representatives of all four septin groups. Among the interactors detected, we found predominantly other septins, confirming the tendency of septins to engage in the formation of homo- and heteropolymeric filaments. Conclusions/Significance: If we take as reference the reported arrangement of the septins 2, 6 and 7 within the heterofilament, (7-6-2-2-6-7), we note that the majority of the observed interactions respect the ""group rule"", i.e. members of the same group (e. g. 6, 8, 10 and 11) can replace each other in the specific position along the heterofilament. Septins of the SEPT6 group preferentially interacted with septins of the SEPT2 group (p<0.001), SEPT3 group (p<0.001) and SEPT7 group (p<0.001). SEPT2 type septins preferentially interacted with septins of the SEPT6 group (p<0.001) aside from being the only septin group which interacted with members of its own group. Finally, septins of the SEPT3 group interacted preferentially with septins of the SEPT7 group (p<0.001). Furthermore, we found non-septin interactors which can be functionally attributed to a variety of different cellular activities, including: ubiquitin/sumoylation cycles, microtubular transport and motor activities, cell division and the cell cycle, cell motility, protein phosphorylation/signaling, endocytosis, and apoptosis.Fundao de Amparo a Pesquisa do Estado Sao Paulo (Fapesp)CAPES: Coordenao de Aperfeioamento de Pessoal de Navel SuperiorConselho Nacional de Pesquisa e Desenvolvimento (CNPq)Laboratorio Nacional de Biociencias-Centro Nacional de Pesquisa em Energia e Materais (LNBio-CNPEM