1,645 research outputs found

    Computational Difficulty of Global Variations in the Density Matrix Renormalization Group

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    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

    Quantum Interactive Proofs with Competing Provers

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    This paper studies quantum refereed games, which are quantum interactive proof systems with two competing provers: one that tries to convince the verifier to accept and the other that tries to convince the verifier to reject. We prove that every language having an ordinary quantum interactive proof system also has a quantum refereed game in which the verifier exchanges just one round of messages with each prover. A key part of our proof is the fact that there exists a single quantum measurement that reliably distinguishes between mixed states chosen arbitrarily from disjoint convex sets having large minimal trace distance from one another. We also show how to reduce the probability of error for some classes of quantum refereed games.Comment: 13 pages, to appear in STACS 200

    Sampling and Representation Complexity of Revenue Maximization

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    We consider (approximate) revenue maximization in auctions where the distribution on input valuations is given via "black box" access to samples from the distribution. We observe that the number of samples required -- the sample complexity -- is tightly related to the representation complexity of an approximately revenue-maximizing auction. Our main results are upper bounds and an exponential lower bound on these complexities

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

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    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 (11/eϵ)(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

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

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    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 ω(G)\omega^*(G) of a game GG 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 nn-fold parallel repetition GnG^n of GG consists of nn instances of GG where the players receive all the inputs at the same time and produce all the outputs at the same time. They win GnG^n if they win each instance of GG. In this paper we show that for any game GG such that ω(G)=1ε<1\omega^*(G) = 1 - \varepsilon < 1, ω(Gn)\omega^*(G^n) decreases exponentially in nn. First, for any game GG on the uniform distribution, we show that ω(Gn)=(1ε2)Ω(nlog(IO)log(ε))\omega^*(G^n) = (1 - \varepsilon^2)^{\Omega\left(\frac{n}{\log(|I||O|)} - |\log(\varepsilon)|\right)}, where I|I| and O|O| are the sizes of the input and output sets. From this result, we show that for any entangled game GG, ω(Gn)(1ε2)Ω(nQlog(IO)log(ε)Q)\omega^*(G^n) \le (1 - \varepsilon^2)^{\Omega(\frac{n}{Q\log(|I||O|)} - \frac{|\log(\varepsilon)|}{Q})} where pp is the input distribution of GG and Q=I2maxxypxy2minxypxyQ= \frac{|I|^2 \max_{xy} p_{xy}^2 }{\min_{xy} p_{xy} }. This implies parallel repetition with exponential decay as long as minxy{pxy}0\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

    Replica Placement on Bounded Treewidth Graphs

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    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(logn)o(\log n). We study the problem in terms of the treewidth tt 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

    AMS measurements of cosmogenic and supernova-ejected radionuclides in deep-sea sediment cores

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    Samples of two deep-sea sediment cores from the Indian Ocean are analyzed with accelerator mass spectrometry (AMS) to search for traces of recent supernova activity around 2 Myr ago. Here, long-lived radionuclides, which are synthesized in massive stars and ejected in supernova explosions, namely 26Al, 53Mn and 60Fe, are extracted from the sediment samples. The cosmogenic isotope 10Be, which is mainly produced in the Earths atmosphere, is analyzed for dating purposes of the marine sediment cores. The first AMS measurement results for 10Be and 26Al are presented, which represent for the first time a detailed study in the time period of 1.7-3.1 Myr with high time resolution. Our first results do not support a significant extraterrestrial signal of 26Al above terrestrial background. However, there is evidence that, like 10Be, 26Al might be a valuable isotope for dating of deep-sea sediment cores for the past few million years.Comment: 5 pages, 2 figures, Proceedings of the Heavy Ion Accelerator Symposium on Fundamental and Applied Science, 2013, will be published by the EPJ Web of conference

    A Characterization of Visibility Graphs for Pseudo-Polygons

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    In this paper, we give a characterization of the visibility graphs of pseudo-polygons. We first identify some key combinatorial properties of pseudo-polygons, and we then give a set of five necessary conditions based off our identified properties. We then prove that these necessary conditions are also sufficient via a reduction to a characterization of vertex-edge visibility graphs given by O'Rourke and Streinu

    Limitations to Frechet's Metric Embedding Method

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    Frechet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Frechet embedding is Bourgain's embedding. The authors have recently shown that for every e>0 any n-point metric space contains a subset of size at least n^(1-e) which embeds into l_2 with distortion O(\log(2/e) /e). The embedding we used is non-Frechet, and the purpose of this note is to show that this is not coincidental. Specifically, for every e>0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Frechet embedding into l_p on subsets of size at least n^{1/2 + e} is \Omega((\log n)^{1/p}).Comment: 10 pages, 1 figur

    Approximating the minimum directed tree cover

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    Given a directed graph GG with non negative cost on the arcs, a directed tree cover of GG is a rooted directed tree such that either head or tail (or both of them) of every arc in GG is touched by TT. The minimum directed tree cover problem (DTCP) is to find a directed tree cover of minimum cost. The problem is known to be NPNP-hard. In this paper, we show that the weighted Set Cover Problem (SCP) is a special case of DTCP. Hence, one can expect at best to approximate DTCP with the same ratio as for SCP. We show that this expectation can be satisfied in some way by designing a purely combinatorial approximation algorithm for the DTCP and proving that the approximation ratio of the algorithm is max{2,ln(D+)}\max\{2, \ln(D^+)\} with D+D^+ is the maximum outgoing degree of the nodes in GG.Comment: 13 page
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