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The article highlights some of the possibilities of the new matrix-matrix approach to solving problems on graphs and hypergraphs. The mivarno-matrix approach is based on the rejection of a complete search and construction of special matrices that allow analyzing "the entire labyrinth of the graph". The approach allows to reduce the computational complexity of algorithms that were considered exhaustive (NP-complete) to quadratic or even linear.

This public document was automatically mirrored from PDFy . Original filename: Quantum Computational Complexity in the Presence of Closed Timelike Curves.pdf URL: https://pdf.yt/d/RD036e3G04KM8qIs Upload date: September 03, 2014 18:57:54

It is well-known that in fluid dynamics an alternative to customary direct solution methods (based on the discretization of the fluid fields) is provided by so-called \emph{particle simulation methods}. Particle simulation methods rely typically on appropriate \emph{kinetic models} for the fluid equations which permit the evaluation of the fluid fields in terms of suitable expectation values (or \emph{momenta}) of the kinetic distribution function $f(\mathbf{r,v},t),$ being respectively $\mathbf{r}$ and\textbf{\}$\mathbf{v}$ the position an velocity of a test particle with probability density $f(\mathbf{r,v},t)$. These kinetic models can be continuous or discrete in phase space, yielding respectively \emph{continuous} or \emph{discrete kinetic models} for the fluids. However, also particle simulation methods may be biased by an undesirable computational complexity. In particular, a fundamental issue is to estimate the algorithmic complexity of numerical simulations based on traditional LBM's (Lattice-Boltzmann methods; for review see Succi, 2001 \cite{Succi}). These methods, based on a discrete kinetic approach, represent currently an interesting alternative to direct solution methods. Here we intend to prove that for incompressible fluids fluids LBM's may present a high complexity. The goal of the investigation is to present a detailed account of the origin of the various complexity sources appearing in customary LBM's. The result is relevant to establish possible strategies for improving the numerical efficiency of existing numerical methods.

Define Minimum Soapy Union (MinSU) as the following optimization problem: given a $k$-tuple $(X_1, X_2,..., X_k)$ of finite integer sets, find a $k$-tuple $(t_1, t_2,..., t_k)$ of integers that minimizes the cardinality of $(X_1 + t_1) \cup (X_2 + t_2) \cup ... \cup (X_n + t_k)$. We show that MinSU is NP-complete, APX-hard, and polynomial for fixed $k$. MinSU appears naturally in the context of protein shotgun sequencing: Here, the protein is cleaved into short and overlapping peptides, which are then analyzed by tandem mass spectrometry. To improve the quality of such spectra, one then asks for the mass of the unknown prefix (the shift) of the spectrum, such that the resulting shifted spectra show a maximum agreement. For real-world data the problem is even more complicated than our definition of MinSU; but our intractability results clearly indicate that it is unlikely to find a polynomial time algorithm for shotgun protein sequencing.

It is shown that there is no standard spiking neural P system that simulates Turing machines with less than exponential time and space overheads. The spiking neural P systems considered here have a constant number of neurons that is independent of the input length. Following this we construct a universal spiking neural P system with exhaustive use of rules that simulates Turing machines in linear time and has only 10 neurons.

Threats on the stability of a financial system may severely affect the functioning of the entire economy, and thus considerable emphasis is placed on the analyzing the cause and effect of such threats. The financial crisis in the current and past decade has shown that one important cause of instability in global markets is the so-called financial contagion, namely the spreading of instabilities or failures of individual components of the network to other, perhaps healthier, components. This leads to a natural question of whether the regulatory authorities could have predicted and perhaps mitigated the current economic crisis by effective computations of some stability measure of the banking networks. Motivated by such observations, we consider the problem of defining and evaluating stabilities of both homogeneous and heterogeneous banking networks against propagation of synchronous idiosyncratic shocks given to a subset of banks. We formalize the homogeneous banking network model of Nier et al. and its corresponding heterogeneous version, formalize the synchronous shock propagation procedures, define two appropriate stability measures and investigate the computational complexities of evaluating these measures for various network topologies and parameters of interest. Our results and proofs also shed some light on the properties of topologies and parameters of the network that may lead to higher or lower stabilities.

It is shown that the canonical problem of classical statistical thermodynamics, the computation of the partition function, is in the case of +/-J Ising spin glasses a particular instance of certain simple sums known as quadratically signed weight enumerators (QWGTs). On the other hand it is known that quantum computing is polynomially equivalent to classical probabilistic computing with an oracle for estimating QWGTs. This suggests a connection between the partition function estimation problem for spin glasses and quantum computation. This connection extends to knots and graph theory via the equivalence of the Kauffman polynomial and the partition function for the Potts model.

Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which represents a valid quantification of resources given the ability to construct compact regions of closed timelike curves. The notion of self-consistent evolution for quantum computers whose components follow closed timelike curves, as pointed out by Deutsch [Phys. Rev. D {\bf 44}, 3197 (1991)], implies that the evolution of the chronology respecting components which interact with the closed timelike curve components is nonlinear. We demonstrate that this nonlinearity can be used to efficiently solve computational problems which are generally thought to be intractable. In particular we demonstrate that a quantum computer which has access to closed timelike curve qubits can solve NP-complete problems with only a polynomial number of quantum gates.

In this paper we study MapReduce computations from a complexity-theoretic perspective. First, we formulate a uniform version of the MRC model of Karloff et al. (2010). We then show that the class of regular languages, and moreover all of sublogarithmic space, lies in constant round MRC. This result also applies to the MPC model of Andoni et al. (2014). In addition, we prove that, conditioned on a variant of the Exponential Time Hypothesis, there are strict hierarchies within MRC so that increasing the number of rounds or the amount of time per processor increases the power of MRC. To the best of our knowledge we are the first to approach the MapReduce model with complexity-theoretic techniques, and our work lays the foundation for further analysis relating MapReduce to established complexity classes.

We study the minimum-concave-cost flow problem on a two-dimensional grid. We characterize the computational complexity of this problem based on the number of rows and columns of the grid, the number of different capacities over all arcs, and the location of sources and sinks. The concave cost over each arc is assumed to be evaluated through an oracle machine, i.e., the oracle machine returns the cost over an arc in a single computational step, given the flow value and the arc index. We propose an algorithm whose running time is polynomial in the number of columns of the grid, for the following cases: (1) the grid has a constant number of rows, a constant number of different capacities over all arcs, and sources and sinks in at most two rows; (2) the grid has two rows and a constant number of different capacities over all arcs connecting rows; (3) the grid has a constant number of rows and all sources in one row, with infinite capacity over each arc. These are presumably the most general polynomially solvable cases, since we show the problem becomes NP-hard when any condition in these cases is removed. Our results apply to abundant variants and generalizations of the dynamic lot sizing model, and answer several questions raised in serial supply chain optimization.

Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or separable, state. This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic (as opposed to randomized) computational complexity. First, I review the one-sided tests for separability, paying particular attention to the semidefinite programming methods. Then, I discuss various ways of formulating the quantum separability problem, from exact to approximate formulations, the latter of which are the paper's main focus. I then give a thorough treatment of the problem's relationship with the complexity classes NP, NP-complete, and co-NP. I also discuss extensions of Gurvits' NP-hardness result to strong NP-hardness of certain related problems. A major open question is whether the NP-contained formulation (QSEP) of the quantum separability problem is Karp-NP-complete; QSEP may be the first natural example of a problem that is Turing-NP-complete but not Karp-NP-complete. Finally, I survey all the proposed (deterministic) algorithms for the quantum separability problem, including the bounded search for symmetric extensions (via semidefinite programming), based on the recent quantum de Finetti theorem; and the entanglement-witness search (via interior-point algorithms and global optimization). These two algorithms have the lowest complexity, with the latter being the best under advice of asymptotically optimal point-coverings of the sphere.

We prove that the word problem of a finitely generated group $G$ is in NP (solvable in polynomial time by a non-deterministic Turing machine) if and only if this group is a subgroup of a finitely presented group $H$ with polynomial isoperimetric function. The embedding can be chosen in such a way that $G$ has bounded distortion in $H$.

This paper studies the computational complexity of disambiguation under probabilistic tree-grammars and context-free grammars. It presents a proof that the following problems are NP-hard: computing the Most Probable Parse (MPP) from a sentence or from a word-graph, and computing the Most Probable Sentence (MPS) from a word-graph. The NP-hardness of computing the MPS from a word-graph also holds for Stochastic Context-Free Grammars. Consequently, the existence of deterministic polynomial-time algorithms for solving these disambiguation problems is a highly improbable event.

While it is in some sense natural that quantum systems are efficient at simulating one another, a less natural question is the efficient simulation of classical systems on quantum computers (QCs). This question was first raised, and partly answered by the PI in the context of Ising spin glasses (an ensemble of classical spin-1/2's with fixed random interactions). Recently, there has been further interest in classical physics simulations on QCs in the context of hydrodynamics, chaos, and knot theory, which has a deep connection to classical statistical mechanics. The purpose of this project is to quantify the computational complexity of the canonical problem of classical statistical mechanics: computation of the classical partition function. We have approached this problem using the Potts model and Ising spin glasses, which are known to give rise to a rich class of hard computational problems. Indeed, instances of spin glass and Potts model problems have been shown to be NP-hard, and have been mapped to problems in graph and knot theory. An instance of the spin glass problem refers here to a particular choice of (i) graph describing the spin glass (spins with q=2 states are located on the vertices, interactions on the edges), and (ii) distribution of interactions. An instance of the Potts model refers to a choice of a graph on whose vertices reside spins with q1 (q an integer) states. In certain limits of particularly simple graphs and distributions these problems are analytically solvable, while in other limits they are computationally hard; hence by tuning the graph and distribution one may expect to traverse a landscape of hardness, whose quantum computational complexity we are exploring.

We describe a web of connections between the following topics: the mathematical theory of voting and social choice; the computational complexity of the Maximum Cut problem; the Gaussian Isoperimetric Inequality and Borell's generalization thereof; the Hypercontractive Inequality of Bonami; and, the analysis of Boolean functions. A major theme is the technique of reducing inequalities about Gaussian functions to inequalities about Boolean functions f : {-1,1}^n -> {-1,1}, and then using induction on n to further reduce to inequalities about functions f : {-1,1} -> {-1,1}. We especially highlight De, Mossel, and Neeman's recent use of this technique to prove the Majority Is Stablest Theorem and Borell's Isoperimetric Inequality simultaneously.

We consider a monopolist seller facing a single buyer with additive valuations over n heterogeneous, independent items. It is known that in this important setting optimal mechanisms may require randomization [HR12], use menus of infinite size [DDT15], and may be computationally intractable [DDT14]. This has sparked recent interest in finding simple mechanisms that obtain reasonable approximations to the optimal revenue [HN12, LY13, BILW14]. In this work we attempt to find the optimal simple mechanism. There are many ways to define simple mechanisms. Here we restrict our search to partition mechanisms, where the seller partitions the items into disjoint bundles and posts a price for each bundle; the buyer is allowed to buy any number of bundles. We give a PTAS for the problem of finding a revenue-maximizing partition mechanism, and prove that the problem is strongly NP-hard. En route, we prove structural properties of near-optimal partition mechanisms which may be of independent interest: for example, there always exists a near-optimal partition mechanism that uses only a constant number of non-trivial bundles (i.e. bundles with more than one item).

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd-Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd-Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd-Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd-Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

Abstract The knapsack problem is an optimization problem in computer science which involves determining the most valuable combination of items that can be packed into a knapsack (a container) with a limited capacity (weight or volume); the goal is to maximize the total profit of the items included in the knapsack without exceeding its capacity. This study extensively analyzes the knapsack problem, exploring the application of three prevalent heuristics: greedy, dynamic programming, and FPTAS algorithms implemented in Python. The study aims to assess how these algorithms perform differently, focusing on program complexity and computational speed. Our main objective is to compare these algorithms and determine the most effective one for solving the knapsack problem as well as to be chosen by the researchers and developers when dealing similar problem in real-world applications. Our methodology involved solving the knapsack problem using the three algorithms within a unified programming environment. We conducted experiments using varying input datasets and recorded the time complexities of the algorithms in each trial. Additionally, we performed Halstead complexity measurements to derive the volume of each algorithm for this study. Subsequently, we compared program complexity in Halstead metrics and computational speed for the three approaches. The research findings reveal that the Greedy algorithm demonstrates superior computational efficiency compared to both Dynamic Programming (D.P) and FPTAS algorithms across various test cases. To advance understanding of the knapsack problem, future research should focus on investigating the performance of other programming languages in addressing combinatorial optimization problems, which would provide valuable insights into language choice impact. Additionally, integrating parallel computing techniques could accelerate solution processes for large-scale problem instances.

One of the fundamental questions of Algorithmic Mechanism Design is whether there exists an inherent clash between truthfulness and computational tractability: in particular, whether polynomial-time truthful mechanisms for combinatorial auctions are provably weaker in terms of approximation ratio than non-truthful ones. This question was very recently answered for universally truthful mechanisms for combinatorial auctions \cite{D11}, and even for truthful-in-expectation mechanisms \cite{DughmiV11}. However, both of these results are based on information-theoretic arguments for valuations given by a value oracle, and leave open the possibility of polynomial-time truthful mechanisms for succinctly described classes of valuations. This paper is the first to prove {\em computational hardness} results for truthful mechanisms for combinatorial auctions with succinctly described valuations. We prove that there is a class of succinctly represented submodular valuations for which no deterministic truthful mechanism provides an $m^{1/2-\epsilon}$-approximation for a constant $\epsilon>0$, unless $NP=RP$ ($m$ denotes the number of items). Furthermore, we prove that even truthful-in-expectation mechanisms cannot approximate combinatorial auctions with certain succinctly described submodular valuations better than within $n^\gamma$, where $n$ is the number of bidders and $\gamma>0$ some absolute constant, unless $NP \subseteq P/poly$. In addition, we prove computational hardness results for two related problems.

The parallel computational complexity or depth of growing network models is investigated. The networks considered are generated by preferential attachment rules where the probability of attaching a new node to an existing node is given by a power, $\alpha$ of the connectivity of the existing node. Algorithms for generating growing networks very quickly in parallel are described and studied. The sublinear and superlinear cases require distinct algorithms. As a result, there is a discontinuous transition in the parallel complexity of sampling these networks corresponding to the discontinuous structural transition at $\alpha=1$, where the networks become scale free. For $\alpha>1$ networks can be generated in constant time while for $0 \leq \alpha < 1$ logarithmic parallel time is required. The results show that these networks have little depth and embody very little history dependence despite being defined by sequential growth rules.

Analyzing the computational complexity of evolutionary algorithms for binary search spaces has significantly increased their theoretical understanding. With this paper, we start the computational complexity analysis of genetic programming. We set up several simplified genetic programming algorithms and analyze them on two separable model problems, ORDER and MAJORITY, each of which captures an important facet of typical genetic programming problems. Both analyses give first rigorous insights on aspects of genetic programming design, highlighting in particular the impact of accepting or rejecting neutral moves and the importance of a local mutation operator.

We propose use of QR factorization with sort and Dijkstra's algorithm for decreasing the computational complexity of the sphere decoder that is used for ML detection of signals on the multi-antenna fading channel. QR factorization with sort decreases the complexity of searching part of the decoder with small increase in the complexity required for preprocessing part of the decoder. Dijkstra's algorithm decreases the complexity of searching part of the decoder with increase in the storage complexity. The computer simulation demonstrates that the complexity of the decoder is reduced by the proposed methods significantly.

Theoretical computer science has found fertile ground in many areas of mathematics. The approach has been to consider classical problems through the prism of computational complexity, where the number of basic computational steps taken to solve a problem is the crucial qualitative parameter. This new approach has led to a sequence of advances, in setting and solving new mathematical challenges as well as in harnessing discrete mathematics to the task of solving real-world problems. In this talk, I will survey the development of modern cryptography -- the mathematics behind secret communications and protocols -- in this light. I will describe the complexity theoretic foundations underlying the cryptographic tasks of encryption, pseudo-randomness number generators and functions, zero knowledge interactive proofs, and multi-party secure protocols. I will attempt to highlight the paradigms and proof techniques which unify these foundations, and which have made their way into the mainstream of complexity theory.

We prove that finding an epsilon-Nash equilibrium in a succinctly representable game with many players is PPAD-hard for constant epsilon. Our proof uses succinct games, i.e. games whose payoff function is represented by a circuit. Our techniques build on a recent query complexity lower bound by Babichenko.

Anomaly detection in massive networks has numerous theoretical and computational challenges, especially as the behavior to be detected becomes small in comparison to the larger network. This presentation focuses on recent results in three key technical areas, specifically geared toward spectral methods for detection. We first discuss recent models for network behavior, and how their structure can be exploited for efficient computation of the principal eigenspace of the graph. In addition to the stochasticity of background activity, a graph of interest may be observed through a noisy or imperfect mechanism, which may hinder the detection process. A few simple noise models are discussed, and we demonstrate the ability to fuse multiple corrupted observations and recover detection performance. Finally, we discuss the challenges in scaling the spectral algorithms to large-scale high-performance computing systems, and present preliminary recommendations to achieve good performance with current parallel eigensolvers.

In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of boundary is given either as a list of pixels, or by a Turing Machine.

We consider some well-known families of two-player, zero-sum, perfect information games that can be viewed as special cases of Shapley's stochastic games. We show that the following tasks are polynomial time equivalent: - Solving simple stochastic games. - Solving stochastic mean-payoff games with rewards and probabilities given in unary. - Solving stochastic mean-payoff games with rewards and probabilities given in binary.

We analyse the resilience of the quantum search algorithm in the presence of quantum noise modelled as trace preserving completely positive maps. We study the influence of noise on computational complexity of the quantum search algorithm. We show that only for small amounts of noise the quantum search algorithm is still more efficient than any classical algorithm.

During the period of the contract, 6/15/86-7/31/89, we have develop: I). A parallel multilevel-multiresolution algorithm for Image Processing and low-level Robot vision tasks, II). A Bayesian/Geometric Framework for 3-D shape estimation from 2-D images, appropriate for object recognition and other Robot tasks III). A procedure for rotation and scale invariant representation (coding) and recognition of textures; a computationally efficient algorithm for estimating Markov Random Fields, IV). We have obtained mathematical results concerning convergence and speed of convergence of computational algorithms such as the annealing algorithm, and have studied mathematically the consistency and asymptotic normality of Maximum Likelihood Estimators for Gibbs distributions. Keywords: Computer vision. (KR)

Presented by The Aspen Center for Physics Scott Aaronson is Schlumberger Chair of Computer Science and Director, Quantum Information Center, Department of Computer Science at The University of Texas at Austin. In summer 2022 he announced he would be working for a year at OpenAI on theoretical foundations of AI safety. Know Aspen. Watch GrassRoots. GrassRoots TV programming is entirely funded by direct contributions from members of the Roaring Fork Valley community, and through grants from the City of Aspen Community Non-Profit Grant Program, Pitkin County Healthy Community Fund, The Town of Snowmass Village and The Aspen Thrift Shop. Founded by citizens of Aspen, Colorado in 1972, GrassRoots TV is the first and oldest community operated and programmed open-source media center and television station in the United States. Virtually all of the programming is created by local residents, occasionally with world renown guests. In partnership with the American Archive of Public Media and the Library of Congress, GrassRoots is currently cataloguing, restoring and digitally archiving 500 of the most at-risk original video programs out of its 45,000 title Community Video Archive. Help sustain this community treasure by donating now! Go to: https://bit.ly/GRTVContribute Don't forget to SUBSCRIBE, HIT LIKE and leave a COMMENT to let us know if you enjoyed this video, it is important to us and the community for you to become part of the conversation. Thanks for tuning in! Download the GrassRoots Community Network iOS and Android apps Subscribe for more videos: https://bit.ly/2Ycpi4P Facebook: https://www.facebook.com/GrassRootsCommunityNetwork/ Twitter: https://twitter.com/grassrootstv Web: http://www.grassrootstv.org/

Morphological analysis requires knowledge of the stems, affixes, combinatory patterns, and spelling-change processes of a natural language. The computational difficulty of the task can be clarified by investigating the computational characteristics of specific models of morphological processing. The use of finite-state machinery in the 'two-level' model by Kimmo Kosken-niemi gives it the appearance of computational efficiency, but closer examination shows the model does not guarantee efficient processing. Reductions of the satisfiability problem show that finding the proper lexical-surface correspondence in a two-level generation or recognition problem can be computationally difficult. However, another source of complexity in the existing algorithms can be sharply reduced by changing the implementation of the dictionary component. A merged dictionary with bit-vectors reduces the numbers of choices among alternative dictionary subdivisions by allowing several subdivisions to be searched at once. (Author)

Morphological analysis requires knowledge of the stems, affixes, combinatory patterns, and spelling-change processes of a natural language. The computational difficulty of the task can be clarified by investigating the computational characteristics of specific models of morphological processing. The use of finite-state machinery in the 'two-level' model by Kimmo Kosken-niemi gives it the appearance of computational efficiency, but closer examination shows the model does not guarantee efficient processing. Reductions of the satisfiability problem show that finding the proper lexical-surface correspondence in a two-level generation or recognition problem can be computationally difficult. However, another source of complexity in the existing algorithms can be sharply reduced by changing the implementation of the dictionary component. A merged dictionary with bit-vectors reduces the numbers of choices among alternative dictionary subdivisions by allowing several subdivisions to be searched at once. (Author)

The Lorentz lattice gas is studied from the perspective of computational complexity theory. It is shown that using massive parallelism, particle trajectories can be simulated in a time that scales logarithmically in the length of the trajectory. This result characterizes the ``logical depth" of the Lorentz lattice gas and allows us to compare it to other models in statistical physics.

The low-temperature driven or thermally activated motion of several condensed matter systems is often modeled by the dynamics of interfaces (co-dimension-1 elastic manifolds) subject to a random potential. Two characteristic quantitative features of the energy landscape of such a many-degree-of-freedom system are the ground-state energy and the magnitude of the energy barriers between given configurations. While the numerical determination of the former can be accomplished in time polynomial in the system size, it is shown here that the problem of determining the latter quantity is NP-complete. Exact computation of barriers is therefore (almost certainly) much more difficult than determining the exact ground states of interfaces.

Complexity theory as practiced by physicists and computational complexity theory as practiced by computer scientists both characterize how difficult it is to solve complex problems. Here it is shown that the parameters of a specific model can be adjusted so that the problem of finding its global energy minimum is extremely sensitive to small changes in the problem statement. This result has implications not only for studies of the physics of random systems but may also lead to new strategies for resolving the well-known P versus NP question in computational complexity theory.

We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show that deciding whether a curve in a metrized PL 3-manifold bounds a surface of area less than a given constant C is NP-hard.

conway game the infernal joke

conway game the infernal joke

Title from cover

Title from cover

Title from cover

In this paper we consider stochastic scheduling models where all relevant data (like processing times, release dates, due dates, etc.) are independent random variables, exponentially distributed. We are interested in the computational complexity of determining optimal policies for these stochastic scheduling models. We give a number of examples of models in which the optimal policies can be determined by polynomial time algorithms while the deterministic counterparts of these models are NP-complete. We also give some examples of stochastic scheduling models for which there exists no polynomial time algorithm if P is not equal NP. (Author)

We show that the problem of finding an optimal stochastic 'blind' controller in a Markov decision process is an NP-hard problem. The corresponding decision problem is NP-hard, in PSPACE, and SQRT-SUM-hard, hence placing it in NP would imply breakthroughs in long-standing open problems in computer science. Our result establishes that the more general problem of stochastic controller optimization in POMDPs is also NP-hard. Nonetheless, we outline a special case that is convex and admits efficient global solutions.

Computational complexity is essential to understanding the properties of black hole horizons. The problem of Alice creating a firewall behind the horizon of Bob's black hole is a problem of computational complexity. In general we find that while creating firewalls is possible, it is extremely difficult and probably impossible for black holes that form in sudden collapse, and then evaporate. On the other hand if the radiation is bottled up then after an exponentially long period of time firewalls may be common. It is possible that gravity will provide tools to study problems of complexity; especially the range of complexity between scrambling and exponential complexity.

The theory of computational complexity focuses on functions and, hence, studies programs whose interactive behavior is reduced to a simple question/answer pattern. We propose a broader theory whose ultimate goal is expressing and analyzing the intrinsic difficulty of fully general interactive behaviors. To this extent, we use standard tools from concurrency theory, including labelled transition systems (formalizing behaviors) and their asynchronous extension (providing causality information). Behaviors are implemented by means of a multiprocessor machine executing CCS-like processes. The resulting theory is shown to be consistent with the classical definitions: when we restrict to functional behaviors (i.e., question/answer patterns), we recover several standard computational complexity classes.

The continuous variable quantum key distribution has been considered to have the potential to provide high secret key rate. However, in present experimental demonstrations, the secret key can be distilled only under very small loss rates. Here, by calculating explicitly the computational complexity with the channel transmission, we show that under high loss rate it is hard to distill the secret key in present continuous variable scheme and one of its advantages, the potential of providing high secret key rate, may therefore be limited.

Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem'' when applied to fermions - causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show, that such a solution is almost certainly unattainable by proving that the sign problem is NP-hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP (nondeterministic polynomial) in polynomial time.