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We consider the problem of ranking a set of OT constraints in a manner consistent with data. We speed up Tesar and Smolensky's RCD algorithm to be linear on the number of constraints. This finds a ranking so each attested form x_i beats or ties a particular competitor y_i. We also generalize RCD so each x_i beats or ties all possible competitors. Alas, this more realistic version of learning has no polynomial algorithm unless P=NP! Indeed, not even generation does. So one cannot improve qualitatively upon brute force: Merely checking that a single (given) ranking is consistent with given forms is coNP-complete if the surface forms are fully observed and Delta_2^p-complete if not. Indeed, OT generation is OptP-complete. As for ranking, determining whether any consistent ranking exists is coNP-hard (but in Delta_2^p) if the forms are fully observed, and Sigma_2^p-complete if not. Finally, we show that generation and ranking are easier in derivational theories: in P, and NP-complete.

This is the second of a series of papers in which we construct an information based general theory of optimal error algorithms and analytic computational complexity and study applications of the general theory. In our first paper we studied a general information' model; here we study an 'iterative information' model. We give a general paradigm, based on the pre-image set of an information operator, for obtaining a lower bound on the error of any algorithm using this information. We show that the order of information provides an upper bound on the order of any algorithm using this information. This upper bound order leads to a lower bound on the complexity index.

This document describes exploratory research on a distributed, trajectory oriented approach for traffic complexity management. The approach is to manage traffic complexity based on preserving trajectory flexibility and minimizing constraints. In particular, the document presents metrics for trajectory flexibility; a method for estimating these metrics based on discrete time and degree of freedom assumptions; a planning algorithm using these metrics to preserve flexibility; and preliminary experiments testing the impact of preserving trajectory flexibility on traffic complexity. The document also describes an early demonstration capability of the trajectory flexibility preservation function in the NASA Autonomous Operations Planner (AOP) platform.

In many assembly tasks, it is necessary to insure the stability of a subcollection of contacting object. To achieve stability, it is often necessary to introduce fixture elements (also called fingers in some work) to help hold objects in place. In this paper, the complexity of stabilizing multiple contacting bodies with the fewest number of fixture elements possible is considered. Standard fixture elements of the type explored in previous single- object grasping work are considered, along with two generalized fixture element variants. The types of stability considered are: form-closure (complete immobility of the assembly); stability with respect to a specific external force and torque on each body; and stability in the neighborhood of a specific external force and torque on each body. The major result is that for most of the combinations of fixture element varieties, and types of stability considered, achieving an optimal solution (that is, finding a smallest set of fixture elements yielding stability) is NP-hard. However, for many fixturing problems it seems likely that suboptimal, yet acceptably small solutions can be found in polynomial time, and some candidate algorithms are presented

We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove that it never differs from topological complexity by more than $1$, thus showing that the latter invariant addresses the problem of the existence of motion planners which are "efficient".

We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove that it never differs from topological complexity by more than $1$, thus showing that the latter invariant addresses the problem of the existence of motion planners which are "efficient".

We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove that it never differs from topological complexity by more than $1$, thus showing that the latter invariant addresses the problem of the existence of motion planners which are "efficient".

We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove that it never differs from topological complexity by more than $1$, thus showing that the latter invariant addresses the problem of the existence of motion planners which are "efficient".

We introduce a new weighted averaging scheme using "Fenchel-type" operators to recover primal solutions in the alternating minimization-type algorithm (AMA) for prototype constrained convex optimization. Our approach combines the classical AMA idea in \cite{Tseng1991} and Nesterov's prox-function smoothing technique without requiring the strong convexity of the objective function. We develop a new non-accelerated primal-dual AMA method and estimate its primal convergence rate both on the objective residual and on the feasibility gap. Then, we incorporate Nesterov's accelerated step into this algorithm and obtain a new accelerated primal-dual AMA variant endowed with a rigorous convergence rate guarantee. We show that the worst-case iteration-complexity in this algorithm is optimal (in the sense of first-oder black-box models), without imposing the full strong convexity assumption on the objective.

We consider a population of interconnected individuals that, with respect to a piece of information, at each time instant can be subdivided into three (time-dependent) categories: agnostics, influenced, and evangelists. A dynamical process of information diffusion evolves among the individuals of the population according to the following rules. Initially, all individuals are agnostic. Then, a set of people is chosen from the outside and convinced to start evangelizing, i.e., to start spreading the information. When a number of evangelists, greater than a given threshold, communicate with a node v, the node v becomes influenced, whereas, as soon as the individual v is contacted by a sufficiently much larger number of evangelists, it is itself converted into an evangelist and consequently it starts spreading the information. The question is: How to choose a bounded cardinality initial set of evangelists so as to maximize the final number of influenced individuals? We prove that the problem is hard to solve, even in an approximate sense. On the positive side, we present exact polynomial time algorithms for trees and complete graphs. For general graphs, we derive exact parameterized algorithms. We also investigate the problem when the objective is to select a minimum number of evangelists capable of influencing the whole network. Our motivations to study these problems come from the areas of Viral Marketing and the analysis of quantitative models of spreading of influence in social networks.

Talk by Marc Wanner - Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithms @QTMLConference

How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing.

Donald Knuth is one of the greatest and most impactful computer scientists and mathematicians ever. He is the recipient in 1974 of the Turing Award, considered the Nobel Prize of computing. He is the author of the multi-volume work, the magnum opus, The Art of Computer Programming. He made several key contributions to the rigorous analysis of the computational complexity of algorithms. He popularized asymptotic notation, that we all affectionately know as the big-O notation. He also created the TeX typesetting which most computer scientists, physicists, mathematicians, and scientists and engineers use to write technical papers and make them look beautiful. This conversation is part of the Artificial Intelligence podcast. This episode is presented by Cash App. Download it & use code "LexPodcast": Cash App (App Store): https://apple.co/2sPrUHe Cash App (Google Play): https://bit.ly/2MlvP5w INFO: Podcast website: https://lexfridman.com/ai Apple Podcasts: https://apple.co/2lwqZIr Spotify: https://spoti.fi/2nEwCF8 RSS: https://lexfridman.com/category/ai/feed/ Full episodes playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4 Clips playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOeciFP3CBCIEElOJeitOr41 EPISODE LINKS: The Art of Computer Programming (book): https://amzn.to/39kxRwB OUTLINE: 0:00 - Introduction 3:45 - IBM 650 7:51 - Geeks 12:29 - Alan Turing 14:26 - My life is a convex combination of english and mathematics 24:00 - Japanese arrow puzzle example 25:42 - Neural networks and machine learning 27:59 - The Art of Computer Programming 36:49 - Combinatorics 39:16 - Writing process 42:10 - Are some days harder than others? 48:36 - What's the "Art" in the Art of Computer Programming 50:21 - Binary (boolean) decision diagram 55:06 - Big-O notation 58:02 - P=NP 1:10:05 - Artificial intelligence 1:13:26 - Ant colonies and human cognition 1:17:11 - God and the Bible 1:24:28 - Reflection on life 1:28:25 - Facing mortality 1:33:40 - TeX and beautiful typography 1:39:23 - How much of the world do we understand? 1:44:17 - Question for God CONNECT: - Subscribe to this YouTube channel - Twitter: https://twitter.com/lexfridman - LinkedIn: https://www.linkedin.com/in/lexfridman - Facebook: https://www.facebook.com/lexfridman - Instagram: https://www.instagram.com/lexfridman - Medium: https://medium.com/@lexfridman - Support on Patreon: https://www.patreon.com/lexfridman Source: https://www.youtube.com/watch?v=2BdBfsXbST8 Uploader: Lex Fridman

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Carrier-grade networks comprise several layers where different protocols coexist. Nowadays, most of these networks have different control planes to manage routing on different layers, leading to a suboptimal use of the network resources and additional operational costs. However, some routers are able to encapsulate, decapsulate and convert protocols and act as a liaison between these layers. A unified control plane would be useful to optimize the use of the network resources and automate the routing configurations. Software-Defined Networking (SDN) based architectures, such as OpenFlow, offer a chance to design such a control plane. One of the most important problems to deal with in this design is the path computation process. Classical path computation algorithms cannot resolve the problem as they do not take into account encapsulations and conversions of protocols. In this paper, we propose algorithms to solve this problem and study several cases: Path computation without bandwidth constraint, under bandwidth constraint and under other Quality of Service constraints. We study the complexity and the scalability of our algorithms and evaluate their performances on real topologies. The results show that they outperform the previous ones proposed in the literature.

A dissertation submitted to the University of Bristol in accordance with the requirements of the degree of Doctor of Philosophy (PhD) in the Faculty of Engineering, Department of Computer Science, July 2009.

This thesis is aimed at determining the worst case asymptotical time complexity behaviour of algorithms for relational operations that work on extensionally or intensionally represented binary relatons. Those relational operations came from a relational language being designed at Naval Postgraduate School. One particular extensional representation technique and two intensional representation techniques are proposed. The above analysis in turn determines the feasibility of implementing a subset of the relational language on conventional architectures. (Author)

How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness probably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R n (the current best algorithm has complexity roughly n 4 and is conjectured to be n 3 ). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing. This is joint work with Luis Rademacher. ©2006 Microsoft Corporation. All rights reserved.

How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness probably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R n (the current best algorithm has complexity roughly n 4 and is conjectured to be n 3 ). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing. This is joint work with Luis Rademacher. ©2006 Microsoft Corporation. All rights reserved.

A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values. In the framework of idempotent mathematics usually constructions and algorithms are more simple with respect to their traditional analogs. We especially examine algorithms of tropical/idempotent mathematics generated by a collection of basic semiring (or semifield) operations and other "good" operations. Every algorithm of this type has an interval version. The complexity of this interval version coincides with the complexity of the initial algorithm. The interval version of an algorithm of this type gives exact interval estimates for the corresponding output data. Algorithms of linear algebra over idempotent and semirings are examined. In this case, basic algorithms are polynomial as well as their interval versions. This situation is very different from the traditional linear algebra, where basic algorithms are polynomial but the corresponding interval versions are NP-hard and interval estimates are not exact.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

In this study, we address the problem of answering queries over a peer-to-peer system of taxonomy-based sources. A taxonomy states subsumption relationships between negation-free DNF formulas on terms and negation-free conjunctions of terms. To the end of laying the foundations of our study, we first consider the centralized case, deriving the complexity of the decision problem and of query evaluation. We conclude by presenting an algorithm that is efficient in data complexity and is based on hypergraphs. More expressive forms of taxonomies are also investigated, which however lead to intractability. We then move to the distributed case, and introduce a logical model of a network of taxonomy-based sources. On such network, a distributed version of the centralized algorithm is then presented, based on a message passing paradigm, and its correctness is proved. We finally discuss optimization issues, and relate our work to the literature.

We consider optimal distributed computation of a given function of distributed data. The input (data) nodes and the sink node that receives the function form a connected network that is described by an undirected weighted network graph. The algorithm to compute the given function is described by a weighted directed acyclic graph and is called the computation graph. An embedding defines the computation communication sequence that obtains the function at the sink. Two kinds of optimal embeddings are sought, the embedding that---(1)~minimizes delay in obtaining function at sink, and (2)~minimizes cost of one instance of computation of function. This abstraction is motivated by three applications---in-network computation over sensor networks, operator placement in distributed databases, and module placement in distributed computing. We first show that obtaining minimum-delay and minimum-cost embeddings are both NP-complete problems and that cost minimization is actually MAX SNP-hard. Next, we consider specific forms of the computation graph for which polynomial time solutions are possible. When the computation graph is a tree, a polynomial time algorithm to obtain the minimum delay embedding is described. Next, for the case when the function is described by a layered graph we describe an algorithm that obtains the minimum cost embedding in polynomial time. This algorithm can also be used to obtain an approximation for delay minimization. We then consider bounded treewidth computation graphs and give an algorithm to obtain the minimum cost embedding in polynomial time.

Most decision and optimization problems encountered in practice fall into one of two categories with respect to any particular solving method or algorithm: either the problem is solved quickly (easy) or else demands an impractically long computational effort (hard). Recent investigations on model classes of problems have shown that some global parameters, such as the ratio between the constraints to be satisfied and the adjustable variables, are good predictors of problem hardness and, moreover, have an effect analogous to thermodynamical parameters, e.g. temperature, in predicting phases in condensed matter physics [Monasson et al., Nature 400 (1999) 133-137]. Here we show that changes in the values of such parameters can be tracked during a run of the algorithm defining a trajectory through the parameter space. Focusing on 3-Satisfiability, a recognized representative of hard problems, we analyze trajectories generated by search algorithms using growth processes statistical physics. These trajectories can cross well defined phases, corresponding to domains of easy or hard instances, and allow to successfully predict the times of resolution.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Integer least-squares problems, concerned with solving a system of equations where the components of the unknown vector are integer-valued, arise in a wide range of applications. In many scenarios the unknown vector is sparse, i.e., a large fraction of its entries are zero. Examples include applications in wireless communications, digital fingerprinting, and array-comparative genomic hybridization systems. Sphere decoding, commonly used for solving integer least-squares problems, can utilize the knowledge about sparsity of the unknown vector to perform computationally efficient search for the solution. In this paper, we formulate and analyze the sparsity-aware sphere decoding algorithm that imposes $\ell_0$-norm constraint on the admissible solution. Analytical expressions for the expected complexity of the algorithm for alphabets typical of sparse channel estimation and source allocation applications are derived and validated through extensive simulations. The results demonstrate superior performance and speed of sparsity-aware sphere decoder compared to the conventional sparsity-unaware sphere decoding algorithm. Moreover, variance of the complexity of the sparsity-aware sphere decoding algorithm for binary alphabets is derived. The search space of the proposed algorithm can be further reduced by imposing lower bounds on the value of the objective function. The algorithm is modified to allow for such a lower bounding technique and simulations illustrating efficacy of the method are presented. Performance of the algorithm is demonstrated in an application to sparse channel estimation, where it is shown that sparsity-aware sphere decoder performs close to theoretical lower limits.

This is the first of a series of papers constructing an information based general theory of optimal errors and analytic computational complexity. Among the applications are such traditionally diverse areas as approximation, boundary-value problems, quadrature, and nonlinear equations in a finite or infinite dimensional space. Traditionally algorithms are often derived by ad hoc criteria. The information based theory rationalizes the synthesis of algorithms by showing how to construct algorithms which minimize or nearly minimize the error. For certain classes of problems it shows how to construct algorithms (linear optimal error algorithms) which enjoy essentially optimal complexity with respect to all possible algorithms. The existence of strongly non-computable problems is demonstrated. In contrast with the gap theorem of recursively computable functions it is shown that every monotonic real function is the complexity of some problem.

This thesis contains an analysis of two computational problems. The first problem is discrete quantum Boolean summation, which is a building block of quantum algorithms for many continuous problems, such as integration, approximation, differential equations, and path integration. The second problem is continuous multivariate Feynman-Kac path integration, which is a special case of path integration. The quantum Boolean summation problem can be solved by the quantum summation (QS) algorithm of Brassard, Hoyer, Mosca and Tapp, which approximates the arithmetic mean of a Boolean function. The author improves the error bound of Brassard et al. for the worst-probabilistic setting. The error bound is sharp. He also presents new sharp error bounds in the average-probabilistic and worst-average settings. His average-probabilistic error bounds prove the optimality of the QS algorithm for a certain choice of its parameters. The study of the worst-average error shows that the QS algorithm is not optimal in this setting; one needs to use a certain number of repetitions to regain its optimality. The multivariate Feynman-Kac path integration problem for smooth multivariate functions suffers from the provable curse of dimensionality in the worst-case deterministic setting (i.e., the minimal number of function evaluations needed to compute an approximation depends exponentially on the number of variables). He shows that, in both the randomized and quantum settings, the curse of dimensionality is vanquished (i.e., the minimal number of function evaluations and/or quantum queries required to compute an approximation depends only polynomially on the reciprocal of the desired accuracy and has a bound independent of the number of variables). The exponents of these polynomials are 2 in the randomized setting and 1 in the quantum setting. These exponents can be lowered at the expense of the dependence on the number of variables.

This thesis contains an analysis of two computational problems. The first problem is discrete quantum Boolean summation, which is a building block of quantum algorithms for many continuous problems, such as integration, approximation, differential equations, and path integration. The second problem is continuous multivariate Feynman-Kac path integration, which is a special case of path integration. The quantum Boolean summation problem can be solved by the quantum summation (QS) algorithm of Brassard, Hoyer, Mosca and Tapp, which approximates the arithmetic mean of a Boolean function. The author improves the error bound of Brassard et al. for the worst-probabilistic setting. The error bound is sharp. He also presents new sharp error bounds in the average-probabilistic and worst-average settings. His average-probabilistic error bounds prove the optimality of the QS algorithm for a certain choice of its parameters. The study of the worst-average error shows that the QS algorithm is not optimal in this setting; one needs to use a certain number of repetitions to regain its optimality. The multivariate Feynman-Kac path integration problem for smooth multivariate functions suffers from the provable curse of dimensionality in the worst-case deterministic setting (i.e., the minimal number of function evaluations needed to compute an approximation depends exponentially on the number of variables). He shows that, in both the randomized and quantum settings, the curse of dimensionality is vanquished (i.e., the minimal number of function evaluations and/or quantum queries required to compute an approximation depends only polynomially on the reciprocal of the desired accuracy and has a bound independent of the number of variables). The exponents of these polynomials are 2 in the randomized setting and 1 in the quantum setting. These exponents can be lowered at the expense of the dependence on the number of variables.

Typestate systems ensure many desirable properties of imperative programs, including initialization of object fields and correct use of stateful library interfaces. Abstract sets with cardinality constraints naturally generalize typestate properties: relationships between the typestates of objects can be expressed as subset and disjointness relations on sets, and elements of sets can be represented as sets of cardinality one. Motivated by these applications, this paper presents new algorithms and new complexity results for constraints on sets and their cardinalities. We study several classes of constraints and demonstrate a trade-off between their expressive power and their complexity. Our first result concerns a quantifier-free fragment of Boolean Algebra with Presburger Arithmetic. We give a nondeterministic polynomial-time algorithm for reducing the satisfiability of sets with symbolic cardinalities to constraints on constant cardinalities, and give a polynomial-space algorithm for the resulting problem. In a quest for more efficient fragments, we identify several subclasses of sets with cardinality constraints whose satisfiability is NP-hard. Finally, we identify a class of constraints that has polynomial-time satisfiability and entailment problems and can serve as a foundation for efficient program analysis.

We consider the problem of learning the structure of Ising models (pairwise binary Markov random fields) from i.i.d. samples. While several methods have been proposed to accomplish this task, their relative merits and limitations remain somewhat obscure. By analyzing a number of concrete examples, we show that low-complexity algorithms often fail when the Markov random field develops long-range correlations. More precisely, this phenomenon appears to be related to the Ising model phase transition (although it does not coincide with it).

Solving multihomogeneous systems, as a wide range of structured algebraic systems occurring frequently in practical problems, is of first importance. Experimentally, solving these systems with Gr\"obner bases algorithms seems to be easier than solving homogeneous systems of the same degree. Nevertheless, the reasons of this behaviour are not clear. In this paper, we focus on bilinear systems (i.e. bihomogeneous systems where all equations have bidegree (1,1)). Our goal is to provide a theoretical explanation of the aforementionned experimental behaviour and to propose new techniques to speed up the Gr\"obner basis computations by using the multihomogeneous structure of those systems. The contributions are theoretical and practical. First, we adapt the classical F5 criterion to avoid reductions to zero which occur when the input is a set of bilinear polynomials. We also prove an explicit form of the Hilbert series of bihomogeneous ideals generated by generic bilinear polynomials and give a new upper bound on the degree of regularity of generic affine bilinear systems. This leads to new complexity bounds for solving bilinear systems. We propose also a variant of the F5 Algorithm dedicated to multihomogeneous systems which exploits a structural property of the Macaulay matrix which occurs on such inputs. Experimental results show that this variant requires less time and memory than the classical homogeneous F5 Algorithm.

Solving multihomogeneous systems, as a wide range of structured algebraic systems occurring frequently in practical problems, is of first importance. Experimentally, solving these systems with Gr\"obner bases algorithms seems to be easier than solving homogeneous systems of the same degree. Nevertheless, the reasons of this behaviour are not clear. In this paper, we focus on bilinear systems (i.e. bihomogeneous systems where all equations have bidegree (1,1)). Our goal is to provide a theoretical explanation of the aforementionned experimental behaviour and to propose new techniques to speed up the Gr\"obner basis computations by using the multihomogeneous structure of those systems. The contributions are theoretical and practical. First, we adapt the classical F5 criterion to avoid reductions to zero which occur when the input is a set of bilinear polynomials. We also prove an explicit form of the Hilbert series of bihomogeneous ideals generated by generic bilinear polynomials and give a new upper bound on the degree of regularity of generic affine bilinear systems. This leads to new complexity bounds for solving bilinear systems. We propose also a variant of the F5 Algorithm dedicated to multihomogeneous systems which exploits a structural property of the Macaulay matrix which occurs on such inputs. Experimental results show that this variant requires less time and memory than the classical homogeneous F5 Algorithm.

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. Define $n$ to be stable if for all $k\ge 0$, we have $\|3^k n\|=\|n\|+3k$. In [7], this author and Zelinsky showed that for any $n$, there exists some $K=K(n)$ such that $3^K n$ is stable; however, the proof there provided no upper bound on $K(n)$ or any way of computing it. In this paper, we describe an algorithm for computing $K(n)$, and thereby also show that the set of stable numbers is a computable set. The algorithm is based on considering the defect of a number, defined by $\delta(n):=\|n\|-3\log_3 n$, building on the methods presented in [3]. As a side benefit, this algorithm also happens to allow fast evaluation of the complexities of powers of $2$; we use it to verify that $\|2^k 3^\ell\|=2k+3\ell$ for $k\le48$ and arbitrary $\ell$ (excluding the case $k=\ell=0$), providing more evidence for the conjecture that $\|2^k 3^\ell\|=2k+3\ell$ whenever $k$ and $\ell$ are not both zero. An implementation of these algorithms in Haskell is available.

Like time complexity models that have significantly contributed to the analysis and development of fast algorithms, energy complexity models for parallel algorithms are desired as crucial means to develop energy efficient algorithms for ubiquitous multicore platforms. Ideal energy complexity models should be validated on real multicore platforms and applicable to a wide range of parallel algorithms. However, existing energy complexity models for parallel algorithms are either theoretical without model validation or algorithm-specific without ability to analyze energy complexity for a wide-range of parallel algorithms. This paper presents a new general validated energy complexity model for parallel (multithreaded) algorithms. The new model abstracts away possible multicore platforms by their static and dynamic energy of computational operations and data access, and derives the energy complexity of a given algorithm from its work, span and I/O complexity. The new model is validated by different sparse matrix vector multiplication (SpMV) algorithms and dense matrix multiplication (matmul) algorithms running on high performance computing (HPC) platforms (e.g., Intel Xeon and Xeon Phi). The new energy complexity model is able to characterize and compare the energy consumption of SpMV and matmul kernels according to three aspects: different algorithms, different input matrix types and different platforms. The prediction of the new model regarding which algorithm consumes more energy with different inputs on different platforms, is confirmed by the experimental results. In order to improve the usability and accuracy of the new model for a wide range of platforms, the platform parameters of ICE model are provided for eleven platforms including HPC, accelerator and embedded platforms.

We present complexity and numerical results for a new asynchronous parallel algorithmic method for the minimization of the sum of a smooth nonconvex function and a convex nonsmooth regularizer, subject to both convex and nonconvex constraints. The proposed method hinges on successive convex approximation techniques and a novel probabilistic model that captures key elements of modern computational architectures and asynchronous implementations in a more faithful way than state-of-the-art models. In the companion paper we provided a detailed description on the probabilistic model and gave convergence results for a diminishing stepsize version of our method. Here, we provide theoretical complexity results for a fixed stepsize version of the method and report extensive numerical comparisons on both convex and nonconvex problems demonstrating the efficiency of our approach.

ADA121995

This is an annotated bibliography of over 300 papers and books on optimal algorithms and analytic complexity covering both the eastern European and the western literature. Each bibliographic item consists of a bibliographic reference, a set of keywords, and a short description. A brief history of the subject is also included. (Author)

It is becoming increasingly apparent that probabilistic approaches can overcome conservatism and computational complexity of the classical worst-case deterministic framework and may lead to designs that are actually safer. In this paper we argue that a comprehensive probabilistic robustness analysis requires a detailed evaluation of the robustness function and we show that such evaluation can be performed with essentially any desired accuracy and confidence using algorithms with complexity linear in the dimension of the uncertainty space. Moreover, we show that the average memory requirements of such algorithms are absolutely bounded and well within the capabilities of today's computers. In addition to efficiency, our approach permits control over statistical sampling error and the error due to discretization of the uncertainty radius. For a specific level of tolerance of the discretization error, our techniques provide an efficiency improvement upon conventional methods which is inversely proportional to the accuracy level; i.e., our algorithms get better as the demands for accuracy increase.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

This paper considers the cooperation between a cognitive system and a primary system where multiple cognitive base stations (CBSs) relay the primary user's (PU) signals in exchange for more opportunity to transmit their own signals. The CBSs use amplify-and-forward (AF) relaying and coordinated beamforming to relay the primary signals and transmit their own signals. The objective is to minimize the overall transmit power of the CBSs given the rate requirements of the PU and the cognitive users (CUs). We show that the relaying matrices have unit rank and perform two functions: Matched filter receive beamforming and transmit beamforming. We then develop two efficient algorithms to find the optimal solution. The first one has linear convergence rate and is suitable for distributed implementation, while the second one enjoys superlinear convergence but requires centralized processing. Further, we derive the beamforming vectors for the linear conventional zero-forcing (CZF) and prior zero-forcing (PZF) schemes, which provide much simpler solutions. Simulation results demonstrate the improvement in terms of outage performance due to the cooperation between the primary and cognitive systems.

The computational complexity of algorithms for Four Dimensional Data Assimilation (4DDA) at NASA's Data Assimilation Office (DAO) is discussed. In 4DDA, observations are assimilated with the output of a dynamical model to generate best-estimates of the states of the system. It is thus a mapping problem, whereby scattered observations are converted into regular accurate maps of wind, temperature, moisture and other variables. The DAO is developing and using 4DDA algorithms that provide these datasets, or analyses, in support of Earth System Science research. Two large-scale algorithms are discussed. The first approach, the Goddard Earth Observing System Data Assimilation System (GEOS DAS), uses an atmospheric general circulation model (GCM) and an observation-space based analysis system, the Physical-space Statistical Analysis System (PSAS). GEOS DAS is very similar to global meteorological weather forecasting data assimilation systems, but is used at NASA for climate research. Systems of this size typically run at between 1 and 20 gigaflop/s. The second approach, the Kalman filter, uses a more consistent algorithm to determine the forecast error covariance matrix than does GEOS DAS. For atmospheric assimilation, the gridded dynamical fields typically have More than 10(exp 6) variables, therefore the full error covariance matrix may be in excess of a teraword. For the Kalman filter this problem can easily scale to petaflop/s proportions. We discuss the computational complexity of GEOS DAS and our implementation of the Kalman filter. We also discuss and quantify some of the technical issues and limitations in developing efficient, in terms of wall clock time, and scalable parallel implementations of the algorithms.

We study the problem of computing the upper bound of the discrete Fr\'{e}chet distance for imprecise input, and prove that the problem is NP-hard. This solves an open problem posed in 2010 by Ahn \emph{et al}. If shortcuts are allowed, we show that the upper bound of the discrete Fr\'{e}chet distance with shortcuts for imprecise input can be computed in polynomial time and we present several efficient algorithms.

Integrating value focused thinking with the shortest path problem results in a unique formulation called the multiobjective average shortest path problem. We prove this is NP-complete for general graphs. For directed acyclic graphs, an efficient algorithm and even faster heuristic are proposed. While the worst case error of the heuristic is proven unbounded, its average performance on random graphs is within 3% of the optimal solution. Additionally, a special case of the more general biobjective average shortest path problem is given, allowing tradeoffs between decreases in arc set cardinality and increases in multiobjective value; the algorithm to solve the average shortest path problem provides all the information needed to solve this more difficult biobjective problem. These concepts are then extended to the minimum cost flow problem creating a new formulation we name the multiobjective average minimum cost flow. This problem is proven NP-complete as well. For directed acyclic graphs, two efficient heuristics are developed, and although we prove the error of any successive average shortest path heuristic is in theory unbounded, they both perform very well on random graphs. Furthermore, we define a general biobjective average minimum cost flow problem. The information from the heuristics can be used to estimate the efficient frontier in a special case of this problem trading off total flow and multiobjective value. Finally, several variants of these two problems are discussed. Proofs are conjectured showing the conditions under which the problems are solvable in polynomial time and when they remain NP-complete.

Presentation slides 

viii, 239 p. : 25 cm

In a wireless system with a large number of distributed nodes, the quality of communication can be greatly improved by pooling the nodes to perform joint transmission/reception. In this paper, we consider the problem of optimally selecting a subset of nodes from potentially a large number of candidates to form a virtual multi-antenna system, while at the same time designing their joint linear transmission strategies. We focus on two specific application scenarios: 1) multiple single antenna transmitters cooperatively transmit to a receiver; 2) a single transmitter transmits to a receiver with the help of a number of cooperative relays. We formulate the joint node selection and beamforming problems as cardinality constrained optimization problems with both discrete variables (used for selecting cooperative nodes) and continuous variables (used for designing beamformers). For each application scenario, we first characterize the computational complexity of the joint optimization problem, and then propose novel semi-definite relaxation (SDR) techniques to obtain approximate solutions. We show that the new SDR algorithms have a guaranteed approximation performance in terms of the gap to global optimality, regardless of channel realizations. The effectiveness of the proposed algorithms is demonstrated via numerical experiments.