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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.

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.

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.

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.

This thesis is aimed at determining the worst case asymptotical time complexity behavior of algorithms for relational operations that work on extensionally or intensionally represented binary relations. Those relational operations come 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.

Compute-and-Forward (C&F) has been proposed as an efficient strategy to reduce the backhaul load for the distributed antenna systems. Finding the optimal coefficients in C&F has commonly been treated as a shortest vector problem (SVP), which is N-P hard. The point of our work and of Sahraei's recent work is that the C&F coefficient problem can be much simpler. Due to the special structure of C&F, some low polynomial complexity optimal algorithms have recently been developed. However these methods can be applied to real valued channels and integer based lattices only. In this paper, we consider the complex valued channel with complex integer based lattices. For the first time, we propose a low polynomial complexity algorithm to find the optimal solution for the complex scenario. Then we propose a simple linear search algorithm which is conceptually suboptimal, however numerical results show that the performance degradation is negligible compared to the optimal method. Both algorithms are suitable for lattices over any algebraic integers, and significantly outperform the lattice reduction algorithm. The complexity of both algorithms are investigated both theoretically and numerically. The results show that our proposed algorithms achieve better performance-complexity trade-offs compared to the existing algorithms.

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".

In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a #P-hard problem. On the other hand we describe an algorithm for this problem which is especially suited for low dimensional (say dimensions at most 12) and for highly-symmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.

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).

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).

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.

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.

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.

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

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.

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 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 project considers the dynamic spectrum management (DSM) problem whereby multiple users sharing a common frequency band must choose their transmit power spectra jointly in response to physical channel conditions including the effects of interference. The goal of the users is to maximize a system-wide utility function (e.g., weighted sum-rate of all users), subject to individual power constraints. The proposed work will focus on a general DSM problem formulation which allows correlated signaling rather than being restricted to the conventional independent orthogonal signaling such as OFDM. The general formulation will exploit the concept of 'interference alignment' which is known to provide substantial rate gain over OFDM signalling for general interference channels. We have successfully analyzed the complexity to characterize the optimal spectrum sharing policies and beamforming strategies in interfering broadcast networks and developed efficient computational methods for optimal resource allocations in such networks.

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.

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.

We propose algorithms for nonparametric sample-based spacial change detection and estimation in large scale sensor networks. We collect random samples containing the location of sensors and their local decisions, and assume that the local decisions can be stimulated or normal , reflecting the local strength of some stimulating agent. Then change in the location of the agent manifests itself by a change in the distribution of stimulated sensors. In this paper, we are aiming at developing a test that, given two collections of samples, can decide whether the distribution generating the samples has changed or not, and give an estimated changed area if a change is indeed detected. The focus of this paper is to reduce the complexity of the detection and estimation algorithm. We propose two fast algorithms with almost linear complexity and analyze their completeness, flexibility and robustness.

The broad purpose of this project was to investigate low-complexity interior point decomposition algorithms for stochastic programming. A specific objective was to evaluate algorithms using test problems arising from useful applications. The important direct results of this project include: (1) a new test problem collection that includes problem instances from a variety of application areas; (2) a new package of C-routines for converting SMPS input data into data structures more suitable for implementing algorithms; (3) a new software package, CPA, for two-stage stochastic linear programs. The test problems and input conversion routines have been developed in a general manner to be useful to other researchers. CPA includes volumetric center algorithms that proved to be successful in our computational evaluations. To the best of our knowledge, CPA is the only software for stochastic programming that includes volumetric center algorithms. Items (1), (2) and (3) are freely accessible over the Internet. The important theoretical results of this project include: (4) a new characterization of convexity-preserving maps; (5) a new coordinate-free foundation for projective spaces; (6) a new geometric characterization of one-dimensional projective spaces; (7) new algorithms for bound-constrained nonlinear optimization. These theoretical results are likely to be useful in computational optimization in general.

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.

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.

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.

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.

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)

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.

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.

The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invariants are parameterised by an integer $r \geq 3$. We resolve the question of complexity for $r=3$ and $r=4$, giving simple proofs that computing Turaev-Viro invariants for $r=3$ is polynomial time, but for $r=4$ is \#P-hard. Moreover, we give an explicit fixed-parameter tractable algorithm for arbitrary $r$, and show through concrete implementation and experimentation that this algorithm is practical---and indeed preferable---to the prior state of the art for real computation.

The inefficiency of the Wardrop equilibrium of nonatomic routing games can be eliminated by placing tolls on the edges of a network so that the socially optimal flow is induced as an equilibrium flow. A solution where the minimum number of edges are tolled may be preferable over others due to its ease of implementation in real networks. In this paper we consider the minimum tollbooth (MINTB) problem, which seeks social optimum inducing tolls with minimum support. We prove for single commodity networks with linear latencies that the problem is NP-hard to approximate within a factor of $1.1377$ through a reduction from the minimum vertex cover problem. Insights from network design motivate us to formulate a new variation of the problem where, in addition to placing tolls, it is allowed to remove unused edges by the social optimum. We prove that this new problem remains NP-hard even for single commodity networks with linear latencies, using a reduction from the partition problem. On the positive side, we give the first exact polynomial solution to the MINTB problem in an important class of graphs---series-parallel graphs. Our algorithm solves MINTB by first tabulating the candidate solutions for subgraphs of the series-parallel network and then combining them optimally.

The computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a non-local annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in O((n/k + log k) n^{2/d}) steps. This is a significant speedup over explicit sequential simulation, which takes O(n^{1+2/d}) time on average. Finally, we show that in one dimension internal DLA can be predicted in O(log n) parallel time, and so is in the complexity class NC.

The computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a non-local annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in O((n/k + log k) n^{2/d}) steps. This is a significant speedup over explicit sequential simulation, which takes O(n^{1+2/d}) time on average. Finally, we show that in one dimension internal DLA can be predicted in O(log n) parallel time, and so is in the complexity class NC.

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.

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

Presentation slides 

viii, 239 p. : 25 cm

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 prove that it is NP-hard for a coalition of two manipulators to compute how to manipulate the Borda voting rule. This resolves one of the last open problems in the computational complexity of manipulating common voting rules. Because of this NP-hardness, we treat computing a manipulation as an approximation problem where we try to minimize the number of manipulators. Based on ideas from bin packing and multiprocessor scheduling, we propose two new approximation methods to compute manipulations of the Borda rule. Experiments show that these methods significantly outperform the previous best known %existing approximation method. We are able to find optimal manipulations in almost all the randomly generated elections tested. Our results suggest that, whilst computing a manipulation of the Borda rule by a coalition is NP-hard, computational complexity may provide only a weak barrier against manipulation in practice.

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.

ADA121995

ADA121995

The project undertook theoretical research in quantum algorithms, complexity of quantum computation, quantum primitives, and quantum communication protocols. In the area of complexity, it compared quantum computation models with classical ones, finding counting complexity classes between BQP and AWPP that are likely different from both. It investigated small-depth quantum circuits (both with and without unbounded fan-in gates such as quantum AND) and found lower and upper bounds on their power and complexity. In the area of new quantum primitives, the project found Hamiltonians for the quantum fan-out gate, based on spin-exchange interactions. In the area of quantum algorithms, the project showed that there are efficient quantum algorithms for various group theoretic problems, for example, group intersection and double coset membership for certain classes of solvable groups. It also found a network of efficient quantum reducibilities between these and other group-theoretic problems. These are the project's successes. The project was unsuccessful in some endeavors. It has so far failed to find natural problems in these intermediate classes between BQP and AWPP, or to isolate the more robust classes among these. It did not find further evidence that BQP does not contain NP. There was no significant progress on quantum communication protocols.

How efficiently can the ground state of a local Hamiltonian be computed? This is a question that lies at the heart of an emerging area called quantum Hamiltonian complexity, that addresses fundamental issues in both quantum complexity theory and condensed matter physics. Of particular importance are 1D Hamiltonians. We give a new combinatorial approach to proving the area law for 1D systems via the detectability lemma, in the process exponentially improving on Hastings' bounds in the frustration free case. We also give an efficient algorithm for finding an MPS approximation to the ground state, in the case of constant bond dimension. Entanglement is a fundamental feature of quantum systems, and understanding its nature is a basic challenge in quantum computation. We study it in a number of basic contexts, including the complexity of parallel repetition of entangled games, and Bell-inequalities distinguishing non-locality versus entanglement. We show how to use entanglement to give a way of generating certifiably random numbers which are provably secure even against a quantum adversary. The method is based on an earlier paper in which we report an implementation of optimal extractors against quantum storage.

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

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 a cellular wireless system, users located at cell edges often suffer significant out-of-cell interference. Assuming each base station is equipped with multiple antennas, we can model this scenario as a multiple-input single-output (MISO) interference channel. In this paper we consider a coordinated beamforming approach whereby multiple base stations jointly optimize their downlink beamforming vectors in order to simultaneously improve the data rates of a given group of cell edge users. Assuming perfect channel knowledge, we formulate this problem as the maximization of a system utility (which balances user fairness and average user rates), subject to individual power constraints at each base station. We show that, for the single carrier case and when the number of antennas at each base station is at least two, the optimal coordinated beamforming problem is NP-hard for both the harmonic mean utility and the proportional fairness utility. For general utilities, we propose a cyclic coordinate descent algorithm, which enables each transmitter to update its beamformer locally with limited information exchange, and establish its global convergence to a stationary point. We illustrate its effectiveness in computer simulations by using the space matched beamformer as a benchmark.

In a cellular wireless system, users located at cell edges often suffer significant out-of-cell interference. Assuming each base station is equipped with multiple antennas, we can model this scenario as a multiple-input single-output (MISO) interference channel. In this paper we consider a coordinated beamforming approach whereby multiple base stations jointly optimize their downlink beamforming vectors in order to simultaneously improve the data rates of a given group of cell edge users. Assuming perfect channel knowledge, we formulate this problem as the maximization of a system utility (which balances user fairness and average user rates), subject to individual power constraints at each base station. We show that, for the single carrier case and when the number of antennas at each base station is at least two, the optimal coordinated beamforming problem is NP-hard for both the harmonic mean utility and the proportional fairness utility. For general utilities, we propose a cyclic coordinate descent algorithm, which enables each transmitter to update its beamformer locally with limited information exchange, and establish its global convergence to a stationary point. We illustrate its effectiveness in computer simulations by using the space matched beamformer as a benchmark.