Computational Difficulty of Global Variations in the Density Matrix Renormalization Group

The density matrix renormalization group (DMRG) approach is arguably the most successful method to numerically find ground states of quantum spin chains. It amounts to iteratively locally optimizing matrix-product states, aiming at better and better approximating the true ground state. To date, both a proof of convergence to the globally best approximation and an assessment of its complexity are lacking. Here we establish a result on the computational complexity of an approximation with matrix-product states: The surprising result is that when one globally optimizes over several sites of local Hamiltonians, avoiding local optima, one encounters in the worst case a computationally difficult NP-hard problem (hard even in approximation). The proof exploits a novel way of relating it to binary quadratic programming. We discuss intriguing ramifications on the difficulty of describing quantum many-body systems.Comment: 5 pages, 1 figure, RevTeX, final versio

The Auto/paracrine regulation of endocrine functions: a history of TGF-β and the adrenal cortex

Aquesta breu revisió vol analitzar quinze anys de recerca que han portat a proposar el factor de creixement transformant beta (TGF-β) com un component endogen auto/paracrí de la regulació de les funcions diferenciades de teixits endocrins, com el còrtex adrenal. El TGF-β reuneix els criteris requerits per aquesta funció, és a dir: a) El pèptid TGF-β està produït per les cèll. ules adrenocorticals i expressat in situ en aquest teixit; b) el TGF-β reprimeix considerablement la capacitat esteroidogènica d'aquestes cèll. ules reprimint els marcadors clau de diferenciació; c) l'hormona sistèmica ACTH modula la resposta de les cèll. ules adrenocorticals al TGF-β; d) la supressió de la síntesi de TGF-β suprimeix la inhibició de l'activitat esteroidogènica d'aquestes cèll. ules. El sistema del TGF-β adrenocortical és, al nostre entendre, el primer circuit regulador (loop) clarament establert en un teixit endocrí. Aquest viatge històric és un tribut al nostre amic José Sáez, que va participar activament en el coneixement d'aquesta història.This brief review is intended as a flash-back spanning over fifteen years of research and finally leading to the proposal that TGF-β could be an endogenous, auto/paracrine component in the regulation of the differentiated functions of an endocrine tissue, i.e., the adrenal cortex. TGF-β meets the criteria required for such a function: (i) the peptide is produced by adrenocortical cells and expressed in the tissue in situ; (ii) TGF-β strikingly down-regulates the steroidogenic capacity of these cells by repressing key differentiation markers; (iii) the systemic hormone ACTH modulates the adrenocortical cells’ responsiveness to TGF-β, and (iv) the suppression of TGF-β synthesis releases the inhibition of the steroidogenic activity of these cells. The TGF-β adrenocortical cell system was, to our knowledge, the first autocrine regulatory loop clearly established in an endocrine tissue. This chronological account is a tribute to our friend, José Sáez, who actively contributed to this history

Parallel Repetition of Entangled Games with Exponential Decay via the Superposed Information Cost

In a two-player game, two cooperating but non communicating players, Alice and Bob, receive inputs taken from a probability distribution. Each of them produces an output and they win the game if they satisfy some predicate on their inputs/outputs. The entangled value $\omega^*(G)$ of a game $G$ is the maximum probability that Alice and Bob can win the game if they are allowed to share an entangled state prior to receiving their inputs. The $n$-fold parallel repetition $G^n$ of $G$ consists of $n$ instances of $G$ where the players receive all the inputs at the same time and produce all the outputs at the same time. They win $G^n$ if they win each instance of $G$. In this paper we show that for any game $G$ such that $\omega^*(G) = 1 - \varepsilon < 1$, $\omega^*(G^n)$ decreases exponentially in $n$. First, for any game $G$ on the uniform distribution, we show that $\omega^*(G^n) = (1 - \varepsilon^2)^{\Omega\left(\frac{n}{\log(|I||O|)} - |\log(\varepsilon)|\right)}$, where $|I|$ and $|O|$ are the sizes of the input and output sets. From this result, we show that for any entangled game $G$, $\omega^*(G^n) \le (1 - \varepsilon^2)^{\Omega(\frac{n}{Q\log(|I||O|)} - \frac{|\log(\varepsilon)|}{Q})}$ where $p$ is the input distribution of $G$ and $Q= \frac{|I|^2 \max_{xy} p_{xy}^2 }{\min_{xy} p_{xy} }$. This implies parallel repetition with exponential decay as long as $\min_{xy} \{p_{xy}\} \neq 0$ for general games. To prove this parallel repetition, we introduce the concept of \emph{Superposed Information Cost} for entangled games which is inspired from the information cost used in communication complexity.Comment: In the first version of this paper we presented a different, stronger Corollary 1 but due to an error in the proof we had to modify it in the second version. This third version is a minor update. We correct some typos and re-introduce a proof accidentally commented out in the second versio

Algorithmic and Hardness Results for the Colorful Components Problems

In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph $G$ such that in the resulting graph $G'$ all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want $G'$ to optimize an objective function, the selection of this function being specific to each problem in the framework. We analyze three objective functions, and thus, three different problems, which are believed to be relevant for the biological applications: minimizing the number of singleton vertices, maximizing the number of edges in the transitive closure, and minimizing the number of connected components. Our main result is a polynomial time algorithm for the first problem. This result disproves the conjecture of Zheng et al. that the problem is $NP$-hard (assuming $P \neq NP$). Then, we show that the second problem is $APX$-hard, thus proving and strengthening the conjecture of Zheng et al. that the problem is $NP$-hard. Finally, we show that the third problem does not admit polynomial time approximation within a factor of $|V|^{1/14 - \epsilon}$ for any $\epsilon > 0$, assuming $P \neq NP$ (or within a factor of $|V|^{1/2 - \epsilon}$, assuming $ZPP \neq NP$).Comment: 18 pages, 3 figure

Maximizing Welfare in Social Networks under a Utility Driven Influence Diffusion Model

Motivated by applications such as viral marketing, the problem of influence maximization (IM) has been extensively studied in the literature. The goal is to select a small number of users to adopt an item such that it results in a large cascade of adoptions by others. Existing works have three key limitations. (1) They do not account for economic considerations of a user in buying/adopting items. (2) Most studies on multiple items focus on competition, with complementary items receiving limited attention. (3) For the network owner, maximizing social welfare is important to ensure customer loyalty, which is not addressed in prior work in the IM literature. In this paper, we address all three limitations and propose a novel model called UIC that combines utility-driven item adoption with influence propagation over networks. Focusing on the mutually complementary setting, we formulate the problem of social welfare maximization in this novel setting. We show that while the objective function is neither submodular nor supermodular, surprisingly a simple greedy allocation algorithm achieves a factor of $(1-1/e-\epsilon)$ of the optimum expected social welfare. We develop \textsf{bundleGRD}, a scalable version of this approximation algorithm, and demonstrate, with comprehensive experiments on real and synthetic datasets, that it significantly outperforms all baselines.Comment: 33 page

Universal Protocols for Information Dissemination Using Emergent Signals

We consider a population of $n$ agents which communicate with each other in a decentralized manner, through random pairwise interactions. One or more agents in the population may act as authoritative sources of information, and the objective of the remaining agents is to obtain information from or about these source agents. We study two basic tasks: broadcasting, in which the agents are to learn the bit-state of an authoritative source which is present in the population, and source detection, in which the agents are required to decide if at least one source agent is present in the population or not.We focus on designing protocols which meet two natural conditions: (1) universality, i.e., independence of population size, and (2) rapid convergence to a correct global state after a reconfiguration, such as a change in the state of a source agent. Our main positive result is to show that both of these constraints can be met. For both the broadcasting problem and the source detection problem, we obtain solutions with a convergence time of $O(\log^2 n)$ rounds, w.h.p., from any starting configuration. The solution to broadcasting is exact, which means that all agents reach the state broadcast by the source, while the solution to source detection admits one-sided error on a $\varepsilon$-fraction of the population (which is unavoidable for this problem). Both protocols are easy to implement in practice and have a compact formulation.Our protocols exploit the properties of self-organizing oscillatory dynamics. On the hardness side, our main structural insight is to prove that any protocol which meets the constraints of universality and of rapid convergence after reconfiguration must display a form of non-stationary behavior (of which oscillatory dynamics are an example). We also observe that the periodicity of the oscillatory behavior of the protocol, when present, must necessarily depend on the number ^\\# X of source agents present in the population. For instance, our protocols inherently rely on the emergence of a signal passing through the population, whose period is \Theta(\log \frac{n}{^\\# X}) rounds for most starting configurations. The design of clocks with tunable frequency may be of independent interest, notably in modeling biological networks

Replica Placement on Bounded Treewidth Graphs

We consider the replica placement problem: given a graph with clients and nodes, place replicas on a minimum set of nodes to serve all the clients; each client is associated with a request and maximum distance that it can travel to get served and there is a maximum limit (capacity) on the amount of request a replica can serve. The problem falls under the general framework of capacitated set covering. It admits an O(\log n)-approximation and it is NP-hard to approximate within a factor of $o(\log n)$. We study the problem in terms of the treewidth $t$ of the graph and present an O(t)-approximation algorithm.Comment: An abridged version of this paper is to appear in the proceedings of WADS'1

On the Design of Cryptographic Primitives

The main objective of this work is twofold. On the one hand, it gives a brief overview of the area of two-party cryptographic protocols. On the other hand, it proposes new schemes and guidelines for improving the practice of robust protocol design. In order to achieve such a double goal, a tour through the descriptions of the two main cryptographic primitives is carried out. Within this survey, some of the most representative algorithms based on the Theory of Finite Fields are provided and new general schemes and specific algorithms based on Graph Theory are proposed

Budget-restricted utility games with ordered strategic decisions

We introduce the concept of budget games. Players choose a set of tasks and each task has a certain demand on every resource in the game. Each resource has a budget. If the budget is not enough to satisfy the sum of all demands, it has to be shared between the tasks. We study strategic budget games, where the budget is shared proportionally. We also consider a variant in which the order of the strategic decisions influences the distribution of the budgets. The complexity of the optimal solution as well as existence, complexity and quality of equilibria are analyzed. Finally, we show that the time an ordered budget game needs to convergence towards an equilibrium may be exponential

Implementation of higher-order absorbing boundary conditions for the Einstein equations

We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition B_L of order L=l yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions B_L with L<l, which include the widely used freezing-Psi_0 boundary condition that imposes the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in Class. Quantum Grav