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We consider the computational complexity of finding a legal black pebbling of a DAG $G=(V,E)$ with minimum cumulative cost. A black pebbling is a sequence $P_0,\ldots, P_t \subseteq V$ of sets of nodes which must satisfy the following properties: $P_0 = \emptyset$ (we start off with no pebbles on $G$), $\mathsf{sinks}(G) \subseteq \bigcup_{j \leq t} P_j$ (every sink node was pebbled at some point) and $\mathsf{parents}\big(P_{i+1}\backslash P_i\big) \subseteq P_i$ (we can only place a new pebble on a node $v$ if all of $v$'s parents had a pebble during the last round). The cumulative cost of a pebbling $P_0,P_1,\ldots, P_t \subseteq V$ is $\mathsf{cc}(P) = | P_1| + \ldots + | P_t|$. The cumulative pebbling cost is an especially important security metric for data-independent memory hard functions, an important primitive for password hashing. Thus, an efficient (approximation) algorithm would be an invaluable tool for the cryptanalysis of password hash functions as it would provide an automated tool to establish tight bounds on the amortized space-time cost of computing the function. We show that such a tool is unlikely to exist. In particular, we prove the following results. (1) It is $\texttt{NP}\mbox{-}\texttt{Hard}$ to find a pebbling minimizing cumulative cost. (2) The natural linear program relaxation for the problem has integrality gap $\tilde{O}(n)$, where $n$ is the number of nodes in $G$. We conjecture that the problem is hard to approximate. (3) We show that a related problem, find the minimum size subset $S\subseteq V$ such that $\textsf{depth}(G-S) \leq d$, is also $\texttt{NP}\mbox{-}\texttt{Hard}$. In fact, under the unique games conjecture there is no $(2-\epsilon)$-approximation algorithm.

An emerging insight is that ground states of symmetry-protected topological orders (SPTO's) possess latent computational complexity in terms of their many-body entanglement. By introducing a fractional symmetry of SPTO, which requires the invariance under 3-colorable symmetries of a lattice, we prove that every renormalization fixed-point state of 2D $(\mathbb{Z}_2)^m$ SPTO with fractional symmetry can be utilized for universal quantum computation using only Pauli measurements, as long as it belongs to a nontrivial 2D SPTO phase. Our infinite family of fixed-point states may serve as a base model to demonstrate the idea of a "quantum computational phase" of matter, whose states share universal computational complexity ubiquitously.

We study the computational complexity of the physical problem of finding vacua of string theory which agree with data, such as the cosmological constant, and show that such problems are typically NP hard. In particular, we prove that in the Bousso-Polchinski model, the problem is NP complete. We discuss the issues this raises and the possibility that, even if we were to find compelling evidence that some vacuum of string theory describes our universe, we might never be able to find that vacuum explicitly. In a companion paper, we apply this point of view to the question of how early cosmology might select a vacuum.

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.

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

We introduce a prime number generator in the form of a stochastic algorithm. The character of such algorithm gives rise to a continuous phase transition which distinguishes a phase where the algorithm is able to reduce the whole system of numbers into primes and a phase where the system reaches a frozen state with low prime density. In this paper we firstly pretend to give a broad characterization of this phase transition, both in terms of analytical and numerical analysis. Critical exponents are calculated, and data collapse is provided. Further on we redefine the model as a search problem, fitting it in the hallmark of computational complexity theory. We suggest that the system belongs to the class NP. The computational cost is maximal around the threshold, as common in many algorithmic phase transitions, revealing the presence of an easy-hard-easy pattern. We finally relate the nature of the phase transition to an average-case classification of the problem.

We show that for any given norm ball or proper cone, weak membership in its dual ball or dual cone is polynomial-time reducible to weak membership in the given ball or cone. A consequence is that the weak membership or membership problem for a ball or cone is NP-hard if and only if the corresponding problem for the dual ball or cone is NP-hard. In a similar vein, we show that computation of the dual norm of a given norm is polynomial-time reducible to computation of the given norm. This extends to convex functions satisfying a polynomial growth condition: for such a given function, computation of its Fenchel dual/conjugate is polynomial-time reducible to computation of the given function. Hence the computation of a norm or a convex function of polynomial-growth is NP-hard if and only if the computation of its dual norm or Fenchel dual is NP-hard. We discuss implications of these results on the weak membership problem for a symmetric convex body and its polar dual, the polynomial approximability of Mahler volume, and the weak membership problem for the epigraph of a convex function with polynomial growth and that of its Fenchel dual.

In three-dimensional computational topology, the theory of normal surfaces is a tool of great theoretical and practical significance. Although this theory typically leads to exponential time algorithms, very little is known about how these algorithms perform in "typical" scenarios, or how far the best known theoretical bounds are from the real worst-case scenarios. Here we study the combinatorial and algebraic complexity of normal surfaces from both the theoretical and experimental viewpoints. Theoretically, we obtain new exponential lower bounds on the worst-case complexities in a variety of settings that are important for practical computation. Experimentally, we study the worst-case and average-case complexities over a comprehensive body of roughly three billion input triangulations. Many of our lower bounds are the first known exponential lower bounds in these settings, and experimental evidence suggests that many of our theoretical lower bounds on worst-case growth rates may indeed be asymptotically tight.

A graph $G$ is embeddable in $\mathbb{R}^d$ if vertices of $G$ can be assigned with points of $\mathbb{R}^d$ in such a way that all pairs of adjacent vertices are at the distance 1. We show that verifying embeddability of a given graph in $\mathbb{R}^d$ is NP-hard in the case $d > 2$ for all reasonable notions of embeddability.

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 take a whirlwind tour of problems and techniques at the boundary of computer science and statistical physics. We start with a brief description of P, NP, and NP-completeness. We then discuss random graphs, including the emergence of the giant component and the k-core, using techniques from branching processes and differential equations. Using these tools as well as the second moment method, we give upper and lower bounds on the critical clause density for random k-SAT. We end with community detection in networks, variational methods, the Bethe free energy, belief propagation, the detectability transition, and the non-backtracking matrix.

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.

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.

Given a matrix $A$ with $n$ rows, a number $k0$ and any arbitrarily small constant $0

For a finite word $w$ of length $n$ and a class of finite automata $\mathcal A$, we study the Kolmogorov structure function $h_w$ for automatic complexity restricted to $\mathcal A$. We propose an approach to computational statistics based on the minimum $p$-value of $h_w(m)$ over $0\le m\le n$. When $\mathcal A$ is the class of all finite automata we give some upper bounds for $h_w$. When $\mathcal A$ consists of automata that detect several success runs in $w$, we give efficient algorithms to compute $h_w$. When $\mathcal A$ consists of automata that detect one success run, we moreover give an efficient algorithm to compute the $p$-values.

In a variety of studies of dynamical systems, the edge of order and chaos has been singled out as a region of complexity. It was suggested by Wolfram, on the basis of qualitative behaviour of cellular automata, that the computational basis for modelling this region is the Universal Turing Machine. In this paper, following a suggestion of Crutchfield, we try to show that the Turing machine model may often be too powerful as a computational model to describe the boundary of order and chaos. In particular we study the region of the first accumulation of period doubling in unimodal and bimodal maps of the interval, from the point of view of language theory. We show that in relation to the ``extended'' Chomsky hierarchy, the relevant computational model in the unimodal case is the nested stack automaton or the related indexed languages, while the bimodal case is modeled by the linear bounded automaton or the related context-sensitive languages.

In this report we study the problem of determining three-dimensional orientations for noisy projections of randomly oriented identical particles. The problem is of central importance in the tomographic reconstruction of the density map of macromolecular complexes from electron microscope images and it has been studied intensively for more than 30 years. We analyze the computational complexity of the orientation problem and show that while several variants of the problem are $NP$-hard, inapproximable and fixed-parameter intractable, some restrictions are polynomial-time approximable within a constant factor or even solvable in logarithmic space. The orientation search problem is formalized as a constrained line arrangement problem that is of independent interest. The negative complexity results give a partial justification for the heuristic methods used in orientation search, and the positive complexity results on the orientation search have some positive implications also to the problem of finding functionally analogous genes. A preliminary version ``The Computational Complexity of Orientation Search in Cryo-Electron Microscopy'' appeared in Proc. ICCS 2004, LNCS 3036, pp. 231--238. Springer-Verlag 2004.

We determine the computational power of preparing Projected Entangled Pair States (PEPS), as well as the complexity of classically simulating them, and generally the complexity of contracting tensor networks. While creating PEPS allows to solve PP problems, the latter two tasks are both proven to be #P-complete. We further show how PEPS can be used to approximate ground states of gapped Hamiltonians, and that creating them is easier than creating arbitrary PEPS. The main tool for our proofs is a duality between PEPS and postselection which allows to use existing results from quantum compexity.

The problem of Distance Edge Labeling is a variant of Distance Vertex Labeling (also known as $L_{2,1}$ labeling) that has been studied for more than twenty years and has many applications, such as frequency assignment. The Distance Edge Labeling problem asks whether the edges of a given graph can be labeled such that the labels of adjacent edges differ by at least two and the labels of edges of distance two differ by at least one. Labels are chosen from the set $\{0,1,\dots,\lambda\}$ for $\lambda$ fixed. We present a full classification of its computational complexity - a dichotomy between the polynomially solvable cases and the remaining cases which are NP-complete. We characterise graphs with $\lambda \le 4$ which leads to a polynomial-time algorithm recognizing the class and we show NP-completeness for $\lambda \ge 5$ by several reductions from Monotone Not All Equal 3-SAT.

This paper addresses the challenge of high computational complexity in video surveillance systems by proposing an efficient model that integrates hybrid machine learning algorithms (HML) for event recognition. Conventional surveillance methods struggle with processing vast amounts of video data in real-time, leading to scalability, and performance issues. Our proposed approach utilizes convolutional neural networks (CNNs) and recurrent neural networks (RNNs) to enhance the accuracy and efficiency of detecting events. By comparing our model with conventional surveillance techniques motion detection, background subtraction, and frame differencing. We demonstrate significant improvements in frame processing time, object detection speed, energy efficiency, and anomaly detection accuracy. The integration of dynamic model scaling and edge computing further optimizes computational resource usage, making our method a scalable and effective solution for real time surveillance needs. This research highlights the potential of machine learning to revolutionize video surveillance, offering insights into developing more intelligent and responsive security systems. The results of your simulation analysis, indicating performance improvements in accuracy by 0.25%, 0.35%, and 0.45% for the motion detection algorithm, background subtraction, and frame differencing respectively, and in real-time data processing by 5.65%, 4.45%, and 6.75% for the motion detection algorithm, background subtraction, and frame differencing respectively, highlight the potential of machine learning to transform video surveillance into a more intelligent and responsive system.

This paper addresses the challenge of high computational complexity in video surveillance systems by proposing an efficient model that integrates hybrid machine learning algorithms (HML) for event recognition. Conventional surveillance methods struggle with processing vast amounts of video data in real-time, leading to scalability, and performance issues. Our proposed approach utilizes convolutional neural networks (CNNs) and recurrent neural networks (RNNs) to enhance the accuracy and efficiency of detecting events. By comparing our model with conventional surveillance techniques motion detection, background subtraction, and frame differencing. We demonstrate significant improvements in frame processing time, object detection speed, energy efficiency, and anomaly detection accuracy. The integration of dynamic model scaling and edge computing further optimizes computational resource usage, making our method a scalable and effective solution for real time surveillance needs. This research highlights the potential of machine learning to revolutionize video surveillance, offering insights into developing more intelligent and responsive security systems. The results of your simulation analysis, indicating performance improvements in accuracy by 0.25%, 0.35%, and 0.45% for the motion detection algorithm, background subtraction, and frame differencing respectively, and in real-time data processing by 5.65%, 4.45%, and 6.75% for the motion detection algorithm, background subtraction, and frame differencing respectively, highlight the potential of machine learning to transform video surveillance into a more intelligent and responsive system.

This paper addresses the challenge of high computational complexity in video surveillance systems by proposing an efficient model that integrates hybrid machine learning algorithms (HML) for event recognition. Conventional surveillance methods struggle with processing vast amounts of video data in real-time, leading to scalability, and performance issues. Our proposed approach utilizes convolutional neural networks (CNNs) and recurrent neural networks (RNNs) to enhance the accuracy and efficiency of detecting events. By comparing our model with conventional surveillance techniques motion detection, background subtraction, and frame differencing. We demonstrate significant improvements in frame processing time, object detection speed, energy efficiency, and anomaly detection accuracy. The integration of dynamic model scaling and edge computing further optimizes computational resource usage, making our method a scalable and effective solution for real time surveillance needs. This research highlights the potential of machine learning to revolutionize video surveillance, offering insights into developing more intelligent and responsive security systems. The results of your simulation analysis, indicating performance improvements in accuracy by 0.25%, 0.35%, and 0.45% for the motion detection algorithm, background subtraction, and frame differencing respectively, and in real-time data processing by 5.65%, 4.45%, and 6.75% for the motion detection algorithm, background subtraction, and frame differencing respectively, highlight the potential of machine learning to transform video surveillance into a more intelligent and responsive system.

Includes bibliographical references (p. 40-41)

Includes bibliographical references (p. 40-41)

The performance of a new heuristic search algorithm is analyzed. The algorithm uses a formal representation that contains enough information to compute the heuristic evaluation function h(n), without requiring a human expert to provide it. The new algorithm is shown to be less efficient than the Dijkstra algorithm, according to the complexity measure number of node expansion. Researchers attempt an interpretation of this strong negative result. Other properties of the new algorithm are discussed in references Valt80, GSoma79, GSValt80. A short discussion of related research, some of which is in progress, concludes the paper. (Author)

Computational Complexity Analysis is a very powerful tool for use in designing algorithms. Unfortunately, the analysis can sometimes be misleading for the range of input with which the algorithm will actually be used. The aim of this paper is to first give an overview of what computational complexity is, and how to analyse it, and to then explain what parts of complexity analysis can cause confusion, and how to avoid these mistakes. The paper also seeks to investigate ways of solving problems faster by sacrificing certain operating requirements.

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.

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.

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.

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/

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.

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.

The task of numerically approximating the solution of an ordinary differential equation initial-value problems is discussed. Two questions are considered: (1) For any given one-step method, what is the complexity of finding an approximate solution with error less than epsilon. (2) Given an infinite sequence of one-step methods of increasing order, how should the method and the step-size be picked so as to minimize the complexity of finding such an approximation. A methodology is described that handles both questions. It is found that within such a sequence of methods, the following hold under very general circumstances: For any epsilon, O &lt; epsilon &lt; 1, there is a unique choice of order and step-size which minimizes the complexity. (2) As epsilon decreases, both the optimal order and the complexity increase monotonically, tending to infinity as epsilon tends to zero. These results are applied to several classes of one-step methods. In doing so, some new Taylor series methods are used that are asymptotically better than Runge-Kutta methods for problems of small dimension. Moreover, it is proven that among all classes of nonlinear Runge-Kutta methods, those due to Brent have the highest order possible.

Research is summarized in the following areas: certified digital signals, factoring and random graphs, compact knapsacks, NP-complete problems, and indices in a finite field GF(q to the m power). (Author)

Mean filters are among the most commonly used filters in the field of digital signal processing in general and image processing in particular. Therefore, the research and improvement of the filter technology will have a positive impact on both the hardware and the software. Our studies in this paper have shown a new implementation solution, which greatly improves computational complexity, thus reducing the time taken for implementation to be superior to that of the original technique. in experimental results in the form of software. In particular, these suggestions can naturally expand into the n-dimensional space.

Mean filters are among the most commonly used filters in the field of digital signal processing in general and image processing in particular. Therefore, the research and improvement of the filter technology will have a positive impact on both the hardware and the software. Our studies in this paper have shown a new implementation solution, which greatly improves computational complexity, thus reducing the time taken for implementation to be superior to that of the original technique. in experimental results in the form of software. In particular, these suggestions can naturally expand into the n-dimensional space.

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.

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.

Harmful Internet hijacking incidents put in evidence how fragile the Border Gateway Protocol (BGP) is, which is used to exchange routing information between Autonomous Systems (ASes). As proved by recent research contributions, even S-BGP, the secure variant of BGP that is being deployed, is not fully able to blunt traffic attraction attacks. Given a traffic flow between two ASes, we study how difficult it is for a malicious AS to devise a strategy for hijacking or intercepting that flow. We show that this problem marks a sharp difference between BGP and S-BGP. Namely, while it is solvable, under reasonable assumptions, in polynomial time for the type of attacks that are usually performed in BGP, it is NP-hard for S-BGP. Our study has several by-products. E.g., we solve a problem left open in the literature, stating when performing a hijacking in S-BGP is equivalent to performing an interception.