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Voter control problems model situations in which an external agent tries toaffect the result of an election by adding or deleting the fewest number of voters. The goal of the agent is to make a specific candidate either win (\emph{constructive} control) or lose (\emph{destructive} control) the election. We study the constructive and destructive voter control problems whenadding and deleting voters have a \emph{combinatorial flavor}: If we add (resp.\ delete) a voter~$v$, we also add (resp.\ delete) a bundle~$\kappa(v) $ of voters that are associated with~$v$. While the bundle~$\kappa(v)$ may have more than one voter, a voter may also be associated with more than one voter. We analyze the computational complexity of the four voter control problems for the Plurality rule. We obtain that, in general, making a candidate lose is computationally easier than making her win. In particular, if the bundling relation is symmetric (i.e.\ $\forall w\colon w \in \kappa(v) \Leftrightarrow v \in \kappa(w) $), and if each voter has at most two voters associated with him, then destructive control is polynomial-time solvable while the constructive variant remains $\NP$-hard. Even if the bundles are disjoint (i.e.\ $\forall w\colon w \in \kappa(v) \Leftrightarrow \kappa(v) = \kappa(w) $), the constructive problem variants remain intractable. Finally, the minimization variant of constructive control by adding voters does not admit an efficient approximation algorithm, unless P=NP.

An important goal of computational linguistics has been to use linguistic theory to guide the construction of computationally efficient real-world natural language processing systems. At first glance, generalized phrase structure grammar (GPSG) appears to be a blessing on two counts. First, the precise formalisms of GPSG might be a direct and transparent guide for parser design and implementation. Second, since GPSG has weak context-free generative power and context-free languages can be parsed in by a wide range of algorithms, GPSG parsers would appear to run in polynomial time. This widely-assumed GPSG efficient parsability result is misleading: here we prove that the universal recognition problem of current GPSG theory is exponential-polynomial time hard, and assuredly intractable. The paper pinpoints sources of complexity (e.g. metarules and the theory of syntactic features) in the current GPSG theory and concludes with some linguistically and computationally motivated restrictions on GPSG.

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.

Richard Karp is a professor at Berkeley and one of the most important figures in the history of theoretical computer science. In 1985, he received the Turing Award for his research in the theory of algorithms, including the development of the Edmonds-Karp algorithm for solving the maximum flow problem on networks, Hopcroft-Karp algorithm for finding maximum cardinality matchings in bipartite graphs, and his landmark paper in complexity theory called \"Reducibility Among Combinatorial Problems\", in which he proved 21 problems to be NP-complete. This paper was probably the most important catalyst in the explosion of interest in the study of NP-completeness

This article is from Journal of Medical Signals and Sensors , volume 3 . Abstract Owing to its simplicity radix-2 is a popular algorithm to implement fast fourier transform. Radix-2p algorithms have the same order of computational complexity as higher radices algorithms, but still retain the simplicity of radix-2. By defining a new concept, twiddle factor template, in this paper, we propose a method for exact calculation of multiplicative complexity for radix-2p algorithms. The methodology is described for radix-2, radix-22 and radix-23 algorithms. Results show that radix-22 and radix-23 have significantly less computational complexity compared with radix-2. Another interesting result is that while the number of complex multiplications in radix-23 algorithm is slightly more than radix-22, the number of real multiplications for radix-23 is less than radix-22. This is because of the twiddle factors in the form of which need less number of real multiplications and are more frequent in radix-23 algorithm.

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 study non-equilibrium steady states (NESS) of spin chains with boundary Markovian dissipation from the computational complexity point of view. We focus on XX chains whose NESS are matrix product operators (MPO), i.e. with coefficients of a tensor operator basis described by transition amplitudes in an auxiliary space. Encoding quantum algorithms in the auxiliary space, we show that estimating expectations of operators, being local in the sense that each acts on disjoint sets of few spins covering all the system, provides the answers of problems at least as hard as, and believed by many computer scientists to be much harder than, those solved by quantum computers. We draw conclusions on the hardness of the above estimations.

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)

This paper proves that arrangement of music is NP-hard when subject to various constraints: avoiding musical dissonance, limiting how many notes can be played simultaneously, and limiting transition speed between chords. These results imply the computational complexity of related musical problems, including musical choreography and rhythm games.

This paper proves that arrangement of music is NP-hard when subject to various constraints: avoiding musical dissonance, limiting how many notes can be played simultaneously, and limiting transition speed between chords. These results imply the computational complexity of related musical problems, including musical choreography and rhythm games.

Consider the following supposedly-simple problem: compute x satisfying x in S, where S is a convex set conveyed by a separation oracle, with no further information (e.g., no bounding ball containing or intersecting S, etc.). Our interest in this problem stems from fundamental issues involving the interplay of (i) the computational complexity of computing a point x in S, (ii) the geometry of S, and (iii) the stability or conditioning of S under perturbation. Under suitable definitions of these terms, we show herein that problem instances with favorable geometry have favorable computational complexity, validating conventional wisdom. We also show a converse of this implication, by showing that there exist problem instances in certain families characterized by unfavorable geometry, that require more computational effort to solve. This in turn leads, under certain assumptions, to a form of equivalence among computational complexity, the geometry of S, and the conditioning of S. Our measures of the geometry of S, relative to a given (reference) point x, are the aspect ratio A = R/r, as well as R and 1/r, where B(x,R) intersection of S contains a ball of radius r. The aspect ratio arises in the analyses of many algorithms for convex problems, and its importance in convex algorithm analysis has been well-known for several decades. However, the terms R and 1/r in our complexity results are a bit counter-intuitive; nevertheless, we show that the computational complexity must involve these terms in addition to the aspect ratio even when the aspect ratio itself is small. This lower-bound complexity analysis relies on simple features of the separation oracle model of conveying S; if we instead assume that S is conveyed by a self-concordant barrier function, then it is an open challenge to prove such complexity lower-bound.

We classify the computational complexity of the popular video games Portal and Portal 2. We isolate individual mechanics of the game and prove NP-hardness, PSPACE-completeness, or (pseudo)polynomiality depending on the specific game mechanics allowed. One of our proofs generalizes to prove NP-hardness of many other video games such as Half-Life 2, Halo, Doom, Elder Scrolls, Fallout, Grand Theft Auto, Left 4 Dead, Mass Effect, Deus Ex, Metal Gear Solid, and Resident Evil. These results build on the established literature on the complexity of video games.

In this paper, we study the physical layer multicasting to multiple co-channel groups in large-scale antenna systems. The users within each group are interested in a common message and different groups have distinct messages. In particular, we aim at designing the precoding vectors solving the so-called quality of service (QoS) and weighted max-min fairness (MMF) problems, assuming that the channel state information is available at the base station (BS). To solve both problems, the baseline approach exploits the semidefinite relaxation (SDR) technique. Considering a BS with $N$ antennas, the SDR complexity is more than $\mathcal{O}(N^{6})$, which prevents its application in large-scale antenna systems. To overcome this issue, we present two new classes of algorithms that, not only have significantly lower computational complexity than existing solutions, but also largely outperform the SDR based methods. Moreover, we present a novel duality between transformed versions of the QoS and the weighted MMF problems. The duality explicitly determines the solution to the weighted MMF problem given the solution to the QoS problem, and vice versa. Numerical results are used to validate the effectiveness of the proposed solutions and to make comparisons with existing alternatives under different operating conditions.

We discuss the computational complexity of random 2D Ising spin glasses, which represent an interesting class of constraint satisfaction problems for black box optimization. Two extremal cases are considered: (1) the +/- J spin glass, and (2) the Gaussian spin glass. We also study a smooth transition between these two extremal cases. The computational complexity of all studied spin glass systems is found to be dominated by rare events of extremely hard spin glass samples. We show that complexity of all studied spin glass systems is closely related to Frechet extremal value distribution. In a hybrid algorithm that combines the hierarchical Bayesian optimization algorithm (hBOA) with a deterministic bit-flip hill climber, the number of steps performed by both the global searcher (hBOA) and the local searcher follow Frechet distributions. Nonetheless, unlike in methods based purely on local search, the parameters of these distributions confirm good scalability of hBOA with local search. We further argue that standard performance measures for optimization algorithms--such as the average number of evaluations until convergence--can be misleading. Finally, our results indicate that for highly multimodal constraint satisfaction problems, such as Ising spin glasses, recombination-based search can provide qualitatively better results than mutation-based search.

The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (``NP-complete'') and those which are likely hard for quantum computers (``QMA-complete'') by indirect methods. That is, they identify problems of comparable worst-case difficulty without directly determining the individual hardness of any given instance. Statistical mechanical methods can be used to complement this classification by directly extracting information about particular families of instances---typically those that involve optimization---by studying random ensembles of them. These pose unusual and interesting (quantum) statistical mechanical questions and the results shed light on the difficulty of problems for large classes of algorithms as well as providing a window on the contrast between typical and worst case complexity. In these lecture notes we present an introduction to this set of ideas with older work on classical satisfiability and recent work on quantum satisfiability as primary examples. We also touch on the connection of computational hardness with the physical notion of glassiness.

This project investigates the relation between computational complexity and the quality of human belief updating.

Consider an undirected graph modeling a social network, where the vertices represent users, and the edges do connections among them. In the competitive diffusion game, each of a number of players chooses a vertex as a seed to propagate his/her opinion, and then it spreads along the edges in the graphs. The objective of every player is to maximize the number of vertices the opinion infects. In this paper, we investigate a computational problem of asking whether a pure Nash equilibrium exists in the competitive diffusion game on unweighed and weighted graphs, and present several negative and positive results. We first prove that the problem is W[1]-hard when parameterized by the number of players even for unweighted graphs. We also show that the problem is NP-hard even for series-parallel graphs with positive integer weights, and is NP-hard even for forests with arbitrary integer weights. Furthermore, we show that the problem for forest of paths with arbitrary weights is solvable in pseudo-polynomial time; and it is solvable in quadratic time if a given graph is unweighted. We also prove that the problem for chain, cochain, and threshold graphs with arbitrary integer weights is solvable in polynomial time.

Speaker: Thorsten Theobald Date: October, 2003

Considering the problem of sampling from the output photon-counting probability distribution of a linear-optical network for input Gaussian states, we obtain results that are of interest from both quantum theory and the computational complexity theory point of view. We derive a general formula for calculating the output probabilities, and by considering input thermal states, we show that the output probabilities are proportional to permanents of positive-semidefinite Hermitian matrices. It is believed that approximating permanents of complex matrices in general is a #P-hard problem. However, we show that these permanents can be approximated with an algorithm in BPP^NP complexity class, as there exists an efficient classical algorithm for sampling from the output probability distribution. We further consider input squeezed-vacuum states and discuss the complexity of sampling from the probability distribution at the output.

We address the issue of reducing the resource required to compute information-theoretic quantum correlation measures like quantum discord and quantum work deficit in two qubits and higher dimensional systems. We show that determination of the quantum correlation measure is possible even if we utilize a restricted set of local measurements. We find that the determination allows us to obtain a closed form of quantum discord and quantum work deficit for several classes of states, with a low error. We show that the computational error caused by the constraint over the complete set of local measurements reduces fast with an increase in the size of the restricted set, implying usefulness of constrained optimization, especially with the increase of dimensions. We perform quantitative analysis to investigate how the error scales with the system size, taking into account a set of plausible constructions of the constrained set. Carrying out a comparative study, we show that the resource required to optimize quantum work deficit is usually higher than that required for quantum discord. We also demonstrate that minimization of quantum discord and quantum work deficit is easier in the case of two-qubit mixed states of fixed ranks and with positive partial transpose in comparison to the corresponding states having non-positive partial transpose. Applying the methodology to quantum spin models, we show that the constrained optimization can be used with advantage in analyzing such systems in quantum information-theoretic language. For bound entangled states, we show that the error is significantly low when the measurements correspond to the spin observables along the three Cartesian coordinates, and thereby we obtain expressions of quantum discord and quantum work deficit for these bound entangled states.

Satisfiability solvers are increasingly playing a key role in software verification, with particularly effective use in the analysis of security vulnerabilities. String processing is a key part of many software applications, such as browsers and web servers. These applications are susceptible to attacks through malicious data received over network. Automated tools for analyzing the security of such applications, thus need to reason about strings. For efficiency reasons, it is desirable to have a solver that treats strings as first-class types. In this paper, we present some theories of strings that are useful in a software security context and analyze the computational complexity of the presented theories. We use this complexity analysis to motivate a byte-blast approach which employs a Boolean encoding of the string constraints to a corresponding Boolean satisfiability problem.

We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis--Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-$b$ expansion of an algebraic irrational real number cannot be generated by a deterministic pushdown automaton. We also confirm an old claim of Cobham proving that such numbers cannot be generated by a tag machine with dilation factor larger than one.

We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis--Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-$b$ expansion of an algebraic irrational real number cannot be generated by a deterministic pushdown automaton. We also confirm an old claim of Cobham proving that such numbers cannot be generated by a tag machine with dilation factor larger than one.

In this paper we examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion limited aggregation and several widely used algorithms for equilibrating the Ising model are shown to be highly sequential; it is unlikely they can be simulated efficiently in parallel. This is in contrast to Mandelbrot percolation that can be simulated in constant parallel time. Our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory. In addition, the results may serve as a guide to simulation physics.

In this paper we examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion limited aggregation and several widely used algorithms for equilibrating the Ising model are shown to be highly sequential; it is unlikely they can be simulated efficiently in parallel. This is in contrast to Mandelbrot percolation that can be simulated in constant parallel time. Our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory. In addition, the results may serve as a guide to simulation physics.

The computational complexity of solving random 3-Satisfiability (3-SAT) problems is investigated. 3-SAT is a representative example of hard computational tasks; it consists in knowing whether a set of alpha N randomly drawn logical constraints involving N Boolean variables can be satisfied altogether or not. Widely used solving procedures, as the Davis-Putnam-Loveland-Logeman (DPLL) algorithm, perform a systematic search for a solution, through a sequence of trials and errors represented by a search tree. In the present study, we identify, using theory and numerical experiments, easy (size of the search tree scaling polynomially with N) and hard (exponential scaling) regimes as a function of the ratio alpha of constraints per variable. The typical complexity is explicitly calculated in the different regimes, in very good agreement with numerical simulations. Our theoretical approach is based on the analysis of the growth of the branches in the search tree under the operation of DPLL. On each branch, the initial 3-SAT problem is dynamically turned into a more generic 2+p-SAT problem, where p and 1-p are the fractions of constraints involving three and two variables respectively. The growth of each branch is monitored by the dynamical evolution of alpha and p and is represented by a trajectory in the static phase diagram of the random 2+p-SAT problem. Depending on whether or not the trajectories cross the boundary between phases, single branches or full trees are generated by DPLL, resulting in easy or hard resolutions.

It is found that the statistical level fluctuations of the AQC 3-SAT problem undergo a transition from a poisson (regular) fluctuation form to a form consistent with the predictions of Random Matrix Theory. We present data which suggests this transition correlates with the computational phase transition in the classical 3-SAT problem. Application to Gaussian Processes and implication for experiment is discussed.

The relative power of several computational models is considered in this thesis. These models are the Turning machine and its multidimensional variant, the random access machine (RAM), the tree machine, and the pointer machine. The basic computational properties of the pointer machine are examined in more detail. For example, time and space hierarchy theorems for pointer machines are presented. Every Turning machine of time complexity t and space complexity s can be simulated by a pointer machine of time complexity O(t) using O(s/log s) nodes. This strengthens a similar result by van Emde Boas (1989). Every alternating pointer machine of time complexity t can be simulated by a deterministic pointer machine using O(t/log t) nodes. Other results concerning nondeterministic and alternating pointer machines are presented. Every tree machine of time complexity t can be simulated on-line by a log-cost RAM of time complexity O((t log t)/log log t).

Quantitative prediction of the structure, properties, and chemical reactivity of molecules by computer programs using first principles electronic structure methods is a valuable complement to experimental studies, because the latter are often difficult or impossible for transient reactive species. Unfortunately, the electronic structure methods which achieve quantitative chemical accuracy, such as those based on coupled cluster theory, are currently limited in applicability to small molecules of less than about ten first row atoms. This is because of the unphysical increase in computational cost with molecular size in standard implementations. This Phase I SBIR research has developed a new from the ground up approach to reformulating these proven electron correlation methods in terms of atom centered spatially localized quantities. The feasibility and formal advantages of the new formulation has been demonstrated in preliminary work based on the simplest electron correlation method, second order perturbation theory. Trial calculations on medium sized molecules indicated that recovery of both absolute electron correlation energy, and also recovery of relative energies across potential surfaces is excellent. The results have shown the potential of these new local correlation methods to allow the study of much larger molecules than can be treated conventionally, with roughly an order of magnitude improvement being possible.

We have developed a digital signature system whose security rests primarily on the existence of a one-way function. Since many one-way functions are known, and since their existence is essential to even conventional authentication systems, the security of the new system is at least as good as in conventional authentication. The security of previously known digital signature systems depends on the difficulty of factoring and related problems and is open to more question. There is a penalty paid for this security in the increased time required to compute a signature, but recent modifications reduced this penalty to an acceptable level. The signature system uses a form of tree authentication, coupled with a one-way hash function to compress a large authentication file into a single number of approximately 100 bits. A patent disclosure has been filed and a paper will be submitted for publication. (KR)

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.

The two-dimensional continuous wavelet transform (CWT) is characterized by a rotation parameter, in addition to the usual translations and dilations. The CWT has been interpreted as space-frequency representation of two-dimensional signals, where the translation corresponds to the position variable, and the inverse of the scale and the rotation, taken together, correspond to the spatial-frequency variable. The integral of the CWT's squared modulus, with respect to all variables, gives the energy of the original signal. Therefore, an integration on a subset of the parameters gives an energy density in the remaining variables. This paper deals with the implementation of the two basic densities, that is, the position (or aspect-angle) and scale-angle densities.

In this paper, as a main theorem, we prove that the decision version of the Frobenius problem is Sigma_2^P-complete under Karp reductions.Given a finite set A of coprime positive integers, we call the greatest integer that cannot be represented as a nonnegative integer combination of A the Frobenius number, and we denote it as g(A). We call a problem of finding g(A) for a given A the Frobenius problem; moreover, we call a problem of determining whether g(A) >= k for a given pair (A, k) the decision version of the Frobenius problem, where A is a finite set of coprime positive integers and k is a positive integer. For the proof, we construct two Karp reductions. First, we reduce a 2-alternating version of the 3-dimensional matching problem, which is known to be Pi_2^P-complete, to a 2-alternating version of the integer knapsack problem. Then, we reduce the variant of the integer knapsack problem to the complement of the decision version of the Frobenius problem. As a corollary, we obtain the main theorem.

Below is a translation from my Russian paper. I added references, unavailable to me in Moscow. Similar results have been also given in [Schnorr Stumpf 75] (see also [Lynch 75]). Earlier relevant work (classical theorems like Compression, Speed-up, etc.) was done in [Tseitin 56, Rabin 59, Hartmanis Stearns 65, Blum 67, Trakhtenbrot 67, Meyer Fischer 72]. I translated only the part with the statement of the results. Instead of the proof part I appended a later (1979, unpublished) proof sketch of a slightly tighter version. The improvement is based on the results of [Meyer Winklmann 78, Sipser 78]. Meyer and Winklmann extended earlier versions to machines with a separate input and working tape, thus allowing complexities smaller than the input length (down to its log). Sipser showed the space-bounded Halting Problem to require only additive constant overhead. The proof in the appendix below employs both advances to extend the original proofs to machines with a fixed alphabet and a separate input and working space. The extension has no (even logarithmic) restrictions on complexity and no overhead (beyond an additive constant). The sketch is very brief and a more detailed exposition is expected later: [Seiferas Meyer].

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.

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.

One of the central problems in quantum mechanics is to determine the ground state properties of a system of electrons interacting via the Coulomb potential. Since its introduction by Hohenberg, Kohn, and Sham, Density Functional Theory (DFT) has become the most widely used and successful method for simulating systems of interacting electrons, making their original work one of the most cited in physics. In this letter, we show that the field of computational complexity imposes fundamental limitations on DFT, as an efficient description of the associated universal functional would allow to solve any problem in the class QMA (the quantum version of NP) and thus particularly any problem in NP in polynomial time. This follows from the fact that finding the ground state energy of the Hubbard model in an external magnetic field is a hard problem even for a quantum computer, while given the universal functional it can be computed efficiently using DFT. This provides a clear illustration how the field of quantum computing is useful even if quantum computers would never be built.

Cyclotomic fast Fourier transforms (CFFTs) are efficient implementations of discrete Fourier transforms over finite fields, which have widespread applications in cryptography and error control codes. They are of great interest because of their low multiplicative and overall complexities. However, their advantages are shown by inspection in the literature, and there is no asymptotic computational complexity analysis for CFFTs. Their high additive complexity also incurs difficulties in hardware implementations. In this paper, we derive the bounds for the multiplicative and additive complexities of CFFTs, respectively. Our results confirm that CFFTs have the smallest multiplicative complexities among all known algorithms while their additive complexities render them asymptotically suboptimal. However, CFFTs remain valuable as they have the smallest overall complexities for most practical lengths. Our additive complexity analysis also leads to a structured addition network, which not only has low complexity but also is suitable for hardware implementations.

A notion of generalized quantifier in computational complexity theory is explored and used to give a unified treatment of leaf language definability, oracle separations, type 2 operators, and circuits with monoidal gates. Relations to Lindstroem quantifiers are pointed out.

Scott Aaronson is a quantum computer scientist. Please support this podcast by checking out our sponsors: - SimpliSafe: https://simplisafe.com/lex and use code LEX to get a free security camera - Eight Sleep: https://www.eightsleep.com/lex and use code LEX to get $200 off - ExpressVPN: https://expressvpn.com/lexpod and use code LexPod to get 3 months free - BetterHelp: https://betterhelp.com/lex and use code LEX to get 10% off EPISODE LINKS: Scott's Blog: https://www.scottaaronson.com/blog/ Our previous episode: https://www.youtube.com/watch?v=uX5t8EivCaM PODCAST INFO: Podcast website: https://lexfridman.com/podcast Apple Podcasts: https://apple.co/2lwqZIr Spotify: https://spoti.fi/2nEwCF8 RSS: https://lexfridman.com/feed/podcast/ YouTube Full Episodes: https://youtube.com/lexfridman YouTube Clips: https://youtube.com/lexclips SUPPORT & CONNECT: - Check out the sponsors

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.

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.

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 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.

In quantum key distribution(QKD), one can use a classical CSS code to distill the final key. However, there is a constraint for the two codes in CSS code and so far it is unknown how to construct a large CSS code efficiently. Here we show that the BDSW method given by Bennett et al can be modified and the error correction and privacy amplification can be done separately with two INDEPENDENT parity matrices. With such a modification, BDSW method can be used to distill the final key without any classical computational complexity. We also apply the method to the case of imperfect source where a small fraction of signals are tagged by Eve.

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.

We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: ``given an x in V, does x belong to X?'' can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, and by a section of the cone of positive semidefinite quadratic forms. We illustrate some of the results on the Traveling Salesman Polytope, an example of a complicated convex body studied in combinatorial optimization.

We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is $s$-chromatic if it is colorable in $s$ colors and any coloring of it uses at least $s$ colors. The forcing chromatic number $F(G)$ of an $s$-chromatic graph $G$ is the smallest number of vertices which must be colored so that, with the restriction that $s$ colors are used, every remaining vertex has its color determined uniquely. We estimate the computational complexity of $F(G)$ relating it to the complexity class US introduced by Blass and Gurevich. We prove that recognizing if $F(G)\le 2$ is US-hard with respect to polynomial-time many-one reductions. Moreover, this problem is coNP-hard even under the promises that $F(G)\le 3$ and $G$ is 3-chromatic. On the other hand, recognizing if $F(G)\le k$, for each constant $k$, is reducible to a problem in US via disjunctive truth-table reduction. Similar results are obtained also for forcing variants of the clique and the domination numbers of a graph.

This review project aims to systematically examine and synthesize the recent advances in computational modeling (CM) of chronic stress (CS) and its relationship to disease progression (DP), with a particular focus on the concept of allostatic load (AL). The overarching goal is to provide a comprehensive overview of how computational approaches have been utilized to elucidate the physiological mechanisms linking chronic stress to disease, identify methodological trends, and highlight gaps in the current literature.