Showing papers on "Approximation algorithm published in 2008"

Optimal algorithms and inapproximability results for every CSP

Abstract: Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result : If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2-CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.

Constrained Clustering: Advances in Algorithms, Theory, and Applications

Abstract: Since the initial work on constrained clustering, there have been numerous advances in methods, applications, and our understanding of the theoretical properties of constraints and constrained clustering algorithms. Bringing these developments together, Constrained Clustering: Advances in Algorithms, Theory, and Applications presents an extensive collection of the latest innovations in clustering data analysis methods that use background knowledge encoded as constraints. Algorithms The first five chapters of this volume investigate advances in the use of instance-level, pairwise constraints for partitional and hierarchical clustering. The book then explores other types of constraints for clustering, including cluster size balancing, minimum cluster size,and cluster-level relational constraints. Theory It also describes variations of the traditional clustering under constraints problem as well as approximation algorithms with helpful performance guarantees. Applications The book ends by applying clustering with constraints to relational data, privacy-preserving data publishing, and video surveillance data. It discusses an interactive visual clustering approach, a distance metric learning approach, existential constraints, and automatically generated constraints. With contributions from industrial researchers and leading academic experts who pioneered the field, this volume delivers thorough coverage of the capabilities and limitations of constrained clustering methods as well as introduces new types of constraints and clustering algorithms.

Randomized consensus algorithms over large scale networks

Abstract: Various randomized consensus algorithms have been proposed in the literature. In some case randomness is due to the choice of a randomized network communication protocol. In other cases, randomness is simply caused by the potential unpredictability of the environment in which the distributed consensus algorithm is implemented. Conditions ensuring the convergence of these algorithms have already been proposed in the literature. As far as the rate of convergence of such algorithms, two approaches can be proposed. One is based on a mean square analysis, while a second is based on the concept of Lyapunov exponent. In this paper, by some concentration results, we prove that the mean square convergence analysis is the right approach when the number of agents is large. Differently from the existing literature, in this paper we do not stick to average preserving algorithms. Instead, we allow to reach consensus at a point which may differ from the average of the initial states. The advantage of such algorithms is that they do not require bidirectional communication among agents and thus they apply to more general contexts. Moreover, in many important contexts it is possible to prove that the displacement from the initial average tends to zero, when the number of agents goes to infinity.

Optimal resource allocation for MIMO ad hoc cognitive radio networks

Abstract: Maximization of the weighted sum-rate of secondary users (SUs) possibly equipped with multiantenna transmitters and receivers is considered in the context of cognitive radio (CR) networks with coexisting primary users (PUs). The total interference power received at the primary receiver is constrained to maintain reliable communication for the PU. An interference channel configuration is considered for ad hoc networking, where the receivers treat the interference from undesired transmitters as noise. Without the CR constraint, a convergent distributed algorithm is developed to obtain (at least) a locally optimal solution. With the CR constraint, a semidistributed algorithm is introduced. An alternative centralized algorithm based on geometric programming and network duality is also developed. Numerical results show the efficacy of the proposed algorithms. The novel approach is flexible to accommodate modifications aiming at interference alignment. However, the stand-alone weighted sum-rate optimal schemes proposed here have merits over interference-alignment alternatives especially for practical SNR values.

Robust Submodular Observation Selection

Abstract: In many applications, one has to actively select among a set of expensive observations before making an informed decision. For example, in environmental monitoring, we want to select locations to measure in order to most effectively predict spatial phenomena. Often, we want to select observations which are robust against a number of possible objective functions. Examples include minimizing the maximum posterior variance in Gaussian Process regression, robust experimental design, and sensor placement for outbreak detection. In this paper, we present the Submodular Saturation algorithm, a simple and efficient algorithm with strong theoretical approximation guarantees for cases where the possible objective functions exhibit submodularity, an intuitive diminishing returns property. Moreover, we prove that better approximation algorithms do not exist unless NP-complete problems admit efficient algorithms. We show how our algorithm can be extended to handle complex cost functions (incorporating non-unit observation cost or communication and path costs). We also show how the algorithm can be used to near-optimally trade off expected-case (e.g., the Mean Square Prediction Error in Gaussian Process regression) and worst-case (e.g., maximum predictive variance) performance. We show that many important machine learning problems fit our robust submodular observation selection formalism, and provide extensive empirical evaluation on several real-world problems. For Gaussian Process regression, our algorithm compares favorably with state-of-the-art heuristics described in the geostatistics literature, while being simpler, faster and providing theoretical guarantees. For robust experimental design, our algorithm performs favorably compared to SDP-based algorithms. c ©2008 Andreas Krause, H. Brendan McMahan, Carlos Guestrin and Anupam Gupta. KRAUSE, MCMAHAN, GUESTRIN AND GUPTA

Parameterized Complexity and Approximation Algorithms

Abstract: Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research.

Efficient Merging and Filtering Algorithms for Approximate String Searches

Abstract: We study the following problem: how to efficiently find in a collection of strings those similar to a given query string? Various similarity functions can be used, such as edit distance, Jaccard similarity, and cosine similarity. This problem is of great interests to a variety of applications that need a high real-time performance, such as data cleaning, query relaxation, and spellchecking. Several algorithms have been proposed based on the idea of merging inverted lists of grams generated from the strings. In this paper we make two contributions. First, we develop several algorithms that can greatly improve the performance of existing algorithms. Second, we study how to integrate existing filtering techniques with these algorithms, and show that they should be used together judiciously, since the way to do the integration can greatly affect the performance. We have conducted experiments on several real data sets to evaluate the proposed techniques.

Fast Counting of Triangles in Large Real Networks without Counting: Algorithms and Laws

Abstract: How can we quickly find the number of triangles in a large graph, without actually counting them? Triangles are important for real world social networks, lying at the heart of the clustering coefficient and of the transitivity ratio. However, straight-forward and even approximate counting algorithms can be slow, trying to execute or approximate the equivalent of a 3-way database join. In this paper, we provide two algorithms, the eigentriangle for counting the total number of triangles in a graph, and the eigentrianglelocal algorithm that gives the count of triangles that contain a desired node. Additional contributions include the following: (a) We show that both algorithms achieve excellent accuracy, with up to sime 1000x faster execution time, on several, real graphs and (b) we discover two new power laws (degree-triangle and triangleparticipation laws) with surprising properties.

Traveling Salesperson Problems for the Dubins Vehicle

Abstract: In this paper, we study minimum-time motion planning and routing problems for the Dubins vehicle, i.e., a nonholonomic vehicle that is constrained to move along planar paths of bounded curvature, without reversing direction. Motivated by autonomous aerial vehicle applications, we consider the traveling salesperson problem for the Dubins vehicle (DTSP): given n points on a plane, what is the shortest Dubins tour through these points, and what is its length? First, we show that the worst-case length of such a tour grows linearly with n and we propose a novel algorithm with performance within a constant factor of the optimum for the worst-case point sets. In doing this, we also obtain an upper bound on the optimal length in the classical point-to-point problem. Second, we study a stochastic version of the DTSP where the n targets are randomly and independently sampled from a uniform distribution. We show that the expected length of such a tour is of order at least n 2/3 and we propose a novel algorithm yielding a solution with length of order n 2/3 with probability one. Third and finally, we study a dynamic version of the DTSP: given a stochastic process that generates target points, is there a policy that guarantees that the number of unvisited points does not diverge over time? If such stable policies exist, what is the minimum expected time that a newly generated target waits before being visited by the vehicle? We propose a novel stabilizing algorithm such that the expected wait time is provably within a constant factor from the optimum.

Approximation Algorithms for Data Placement Problems

Abstract: We develop approximation algorithms for the problem of placing replicated data in arbitrary networks, where the nodes may both issue requests for data objects and have capacity for storing data objects so as to minimize the average data-access cost. We introduce the data placement problem to model this problem. We have a set of caches $\mathcal{F}$, a set of clients $\mathcal{D}$, and a set of data objects $\mathcal{O}$. Each cache $i$ can store at most $u_i$ data objects. Each client $j\in\mathcal{D}$ has demand $d_j$ for a specific data object $o(j)\in\mathcal{O}$ and has to be assigned to a cache that stores that object. Storing an object $o$ in cache $i$ incurs a storage cost of $f_i^o$, and assigning client $j$ to cache $i$ incurs an access cost of $d_jc_{ij}$. The goal is to find a placement of the data objects to caches respecting the capacity constraints, and an assignment of clients to caches so as to minimize the total storage and client access costs. We present a 10-approximation algorithm for this problem. Our algorithm is based on rounding an optimal solution to a natural linear-programming relaxation of the problem. One of the main technical challenges encountered during rounding is to preserve the cache capacities while incurring only a constant-factor increase in the solution cost. We also introduce the connected data placement problem to capture settings where write-requests are also issued for data objects, so that one requires a mechanism to maintain consistency of data. We model this by requiring that all caches containing a given object be connected by a Steiner tree to a root for that object, which issues a multicast message upon a write to (any copy of) that object. The total cost now includes the cost of these Steiner trees. We devise a 14-approximation algorithm for this problem. We show that our algorithms can be adapted to handle two variants of the problem: (a) a $k$-median variant, where there is a specified bound on the number of caches that may contain a given object, and (b) a generalization where objects have lengths and the total length of the objects stored in any cache must not exceed its capacity.

Better approximation of betweenness centrality

Abstract: Estimating the importance or centrality of the nodes in large networks has recently attracted increased interest. Betweenness is one of the most important centrality indices, which basically counts the number of shortest paths going through a node. Betweenness has been used in diverse applications, e.g., social network analysis or route planning. Since exact computation is prohibitive for large networks, approximation algorithms are important. In this paper, we propose a framework for unbiased approximation of betweenness that generalizes a previous approach by Brandes. Our best new schemes yield significantly better approximation than before for many real world inputs. In particular, we also get good approximations for the betweenness of unimportant nodes.

Max-Product for Maximum Weight Matching: Convergence, Correctness, and LP Duality

Abstract: Max-product "belief propagation" (BP) is an iterative, message-passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization, which involve graphs with many cycles, theoretical results about both the correctness and convergence of the algorithm are known in only a few cases (see section I for references). In this paper, we prove the correctness and convergence of max-product for finding the maximum weight matching (MWM) in bipartite graphs. Even though the underlying graph of the MWM problem has many cycles, somewhat surprisingly we show that the max-product algorithm converges to the correct MWM as long as the MWM is unique. We provide a bound on the number of iterations required and show that for a graph of size n, the computational cost of the algorithm scales as O(n3), which is the same as the computational cost of the best known algorithms for finding the MWM. We also provide an interesting relation between the dynamics of the max-product algorithm and the auction algorithm, which is a well-known distributed algorithm for solving the MWM problem.

Joint multi-label multi-instance learning for image classification

Abstract: In real world, an image is usually associated with multiple labels which are characterized by different regions in the image. Thus image classification is naturally posed as both a multi-label learning and multi-instance learning problem. Different from existing research which has considered these two problems separately, we propose an integrated multi- label multi-instance learning (MLMIL) approach based on hidden conditional random fields (HCRFs), which simultaneously captures both the connections between semantic labels and regions, and the correlations among the labels in a single formulation. We apply this MLMIL framework to image classification and report superior performance compared to key existing approaches over the MSR Cambridge (MSRC) and Corel data sets.

Market equilibrium via a primal--dual algorithm for a convex program

Abstract: We give the first polynomial time algorithm for exactly computing an equilibrium for the linear utilities case of the market model defined by Fisher. Our algorithm uses the primal--dual paradigm in the enhanced setting of KKT conditions and convex programs. We pinpoint the added difficulty raised by this setting and the manner in which our algorithm circumvents it.

Approximation and learning by greedy algorithms

Abstract: We consider the problem of approximating a given element f from a Hilbert space $\mathcal{H}$ by means of greedy algorithms and the application of such procedures to the regression problem in statistical learning theory. We improve on the existing theory of convergence rates for both the orthogonal greedy algorithm and the relaxed greedy algorithm, as well as for the forward stepwise projection algorithm. For all these algorithms, we prove convergence results for a variety of function classes and not simply those that are related to the convex hull of the dictionary. We then show how these bounds for convergence rates lead to a new theory for the performance of greedy algorithms in learning. In particular, we build upon the results in [IEEE Trans. Inform. Theory 42 (1996) 2118–2132] to construct learning algorithms based on greedy approximations which are universally consistent and provide provable convergence rates for large classes of functions. The use of greedy algorithms in the context of learning is very appealing since it greatly reduces the computational burden when compared with standard model selection using general dictionaries.

Improved Approximation Algorithms for Minimum Weight Vertex Separators

Abstract: We develop the algorithmic theory of vertex separators and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into $L_1$ (and even Euclidean embeddings) are insufficient but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an $O(\sqrt{\log n})$ approximation for minimum ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be $\Theta(\sqrt{\log n})$. We also prove an optimal $O(\log k)$-approximate max-flow/min-vertex-cut theorem for arbitrary vertex-capacitated multicommodity flow instances on $k$ terminals. For uniform instances on any excluded-minor family of graphs, we improve this to $O(1)$, and this yields a constant-factor approximation for minimum ratio vertex cuts in such graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best known ratio was $O(\log n)$. These results have a number of applications. We exhibit an $O(\sqrt{\log n})$ pseudoapproximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of $O(\sqrt{\log {opt}})$, where ${opt}$ is the size of an optimal separator, improving over the previous best bound of $O(\log {opt})$. Likewise, we obtain improved approximation ratios for treewidth: in any graph of treewidth $k$, we show how to find a tree decomposition of width at most $O(k \sqrt{\log k})$, whereas previous algorithms yielded $O(k \log k)$. For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constant-factor approximation for the treewidth. This in turn can be used to obtain polynomial-time approximation schemes for several problems in such graphs.

Algorithms for subset selection in linear regression

Abstract: We study the problem of selecting a subset of k random variables to observe that will yield the best linear prediction of another variable of interest, given the pairwise correlations between the observation variables and the predictor variable. Under approximation preserving reductions, this problem is equivalent to the "sparse approximation" problem of approximating signals concisely. The subset selection problem is NP-hard in general; in this paper, we propose and analyze exact and approximation algorithms for several special cases of practical interest. Specifically, we give an FPTAS when the covariance matrix has constant bandwidth, and exact algorithms when the associated covariance graph, consisting of edges for pairs of variables with non-zero correlation, forms a tree or has a large (known) independent set. Furthermore, we give an exact algorithm when the variables can be embedded into a line such that the covariance decreases exponentially in the distance, and a constant-factor approximation when the variables have no "conditional suppressor variables". Much of our reasoning is based on perturbation results for the R2 multiple correlation measure, which is frequently used as a natural measure for "goodness-of-fit statistics". It lies at the core of our FPTAS, and also allows us to extend our exact algorithms to approximation algorithms when the matrix "nearly" falls into one of the above classes. We also use our perturbation analysis to prove approximation guarantees for the widely used "Forward Regression" heuristic under the assumption that the observation variables are nearly independent.

Sieve Algorithms for the Shortest Vector Problem are Practical

Abstract: The most famous lattice problem is the Shortest Vector Problem (SVP), which has many applications in cryptology. The best approximation algorithms known for SVP in high dimension rely on a subroutine for exact SVP in low dimension. In this paper, we assess the practicality of the best (theoretical) algorithm known for exact SVP in low dimension: the sieve algorithm proposed by Ajtai, Kumar and Sivakumar (AKS) in 2001. AKS is a randomized algorithm of time and space complexity 2 O(n) , which is theoretically much lower than the super-exponential complexity of all alternative SVP algorithms. Surprisingly, no implementation and no practical analysis of AKS has ever been reported. It was in fact widely believed that AKS was impractical: for instance, Schnorr claimed in 2003 that the constant hidden in the 2 O(n) complexity was at least 30. In this paper, we show that AKS can actually be made practical: we present a heuristic variant of AKS whose running time is (4/3+! ) n polynomial-time operations, and whose space requirement is(4/3+! ) n/2 polynomially many bits. Our implementation can experimentally find shortest lattice vectors up to dimension 50, but is slower than classical alternative SVP algorithms in these dimensions.

Proportional Fairness in Multi-Rate Wireless LANs

Abstract: In multi-rate wireless LANs, throughput-based fair bandwidth allocation can lead to drastically reduced aggregate throughput. To balance aggregate throughput while serving users in a fair manner, proportional fair or time-based fair scheduling has been proposed to apply at each access point (AP). However, since a realistic deployment of wireless LANs can consist of a network of APs, this paper considers proportional fairness in this much wider setting. Our technique is to intelligently associate users with APs to achieve optimal proportional fairness in a network of APs. We propose two approximation algorithms for periodical offline optimization. Our algorithms are the first approximation algorithms in the literature with a tight worst-case guarantee for the NP-hard problem. Our simulation results demonstrate that our algorithms can obtain an aggregate throughput which can be as much as 2.3 times more than that of the max-min fair allocation in 802.11b. While maintaining aggregate throughput, our approximation algorithms outperform the default user-AP association method in the 802.11b standard significantly in terms of fairness.

Fast Analysis of Large Antenna Arrays Using the Characteristic Basis Function Method and the Adaptive Cross Approximation Algorithm

Abstract: The characteristic basis function method (CBFM) has been hybridized with the adaptive cross approximation (ACA) algorithm to construct a reduced matrix equation in a time-efficient manner and to solve electrically large antenna array problems in-core, with a solve time orders of magnitude less than those in the conventional methods. Various numerical examples are presented that demonstrate that the proposed method has a very good accuracy, computational efficiency and reduced memory storage requirement. Specifically, we analyze large 1-D and 2-D arrays of electrically interconnected tapered slot antennas (TSAs). The entire computational domain is subdivided into many smaller subdomains, each of which supports a set of characteristic basis functions (CBFs). We also present a novel scheme for generating the CBFs that do not conform to the edge condition at the truncated edge of each subdomain, and provide a minor overlap between the CBFs in adjacent subdomains. As a result, the CBFs preserve the continuity of the surface current across the subdomain interfaces, thereby circumventing the need to use separate ldquoconnectionrdquo basis functions.

Discrete laplace operator on meshed surfaces

Abstract: In recent years a considerable amount of work in graphics and geometric optimization used tools based on the Laplace-Beltrami operator on a surface. The applications of the Laplacian include mesh editing, surface smoothing, and shape interpolations among others. However, it has been shown [13, 24, 26] that the popular cotangent approximation schemes do not provide convergent point-wise (or even L2) estimates, while many applications rely on point-wise estimation. Existence of such schemes has been an open question [13].In this paper we propose the first algorithm for approximating the Laplace operator of a surface from a mesh with point-wise convergence guarantees applicable to arbitrary meshed surfaces. We show that for a sufficiently fine mesh over an arbitrary surface, our mesh Laplacian is close to the Laplace-Beltrami operator on the surface at every point of the surface.Moreover, the proposed algorithm is simple and easily implementable. Experimental evidence shows that our algorithm exhibits convergence empirically and compares favorably with cotangentbased methods in providing accurate approximation of the Laplace operator for various meshes.

Constant-Time Approximation Algorithms via Local Improvements

Abstract: We present a technique for transforming classical approximation algorithms into constant-time algorithms that approximate the size of the optimal solution. Our technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. The technique is based on greedily considering local improvements in random order.The problems amenable to our technique include vertex cover, maximum matching, maximum weight matching, set cover, and minimum dominating set. For example, for maximum matching, we give the first constant-time algorithm that for the class of graphs of degree bounded by d, computes the maximum matching size to within epsivn, for any epsivn > 0, where n is the number of nodes in the graph. The running time of the algorithm is independent of n, and only depends on d and epsiv.

Lifetime maximization for connected target coverage in wireless sensor networks

Abstract: In this paper, we consider the connected target coverage (CTC) problem with the objective of maximizing the network lifetime by scheduling sensors into multiple sets, each of which can maintain both target coverage and connectivity among all the active sensors and the sink. We model the CTC problem as a maximum cover tree (MCT) problem and prove that the MCT problem is NP-Complete. We determine an upper bound on the network lifetime for the MCT problem and then develop a (1+w)H(M circ) approximation algorithm to solve it, where w is an arbitrarily small number, H(M circ)=1 lesilesM circ(1/i) and M circ is the maximum number of targets in the sensing area of any sensor. As the protocol cost of the approximation algorithm may be high in practice, we develop a faster heuristic algorithm based on the approximation algorithm called Communication Weighted Greedy Cover (CWGC) algorithm and present a distributed implementation of the heuristic algorithm. We study the performance of the approximation algorithm and CWGC algorithm by comparing them with the lifetime upper bound and other basic algorithms that consider the coverage and connectivity problems independently. Simulation results show that the approximation algorithm and CWGC algorithm perform much better than others in terms of the network lifetime and the performance improvement can be up to 45% than the best-known basic algorithm. The lifetime obtained by our algorithms is close to the upper bound. Compared with the approximation algorithm, the CWGC algorithm can achieve a similar performance in terms of the network lifetime with a lower protocol cost.

Efficient gathering of correlated data in sensor networks

Abstract: In this article, we design techniques that exploit data correlations in sensor data to minimize communication costs (and hence, energy costs) incurred during data gathering in a sensor network. Our proposed approach is to select a small subset of sensor nodes that may be sufficient to reconstruct data for the entire sensor network. Then, during data gathering only the selected sensors need to be involved in communication. The selected set of sensors must also be connected, since they need to relay data to the data-gathering node. We define the problem of selecting such a set of sensors as the connected correlation-dominating set problem, and formulate it in terms of an appropriately defined correlation structure that captures general data correlations in a sensor network.We develop a set of energy-efficient distributed algorithms and competitive centralized heuristics to select a connected correlation-dominating set of small size. The designed distributed algorithms can be implemented in an asynchronous communication model, and can tolerate message losses. We also design an exponential (but nonexhaustive) centralized approximation algorithm that returns a solution within O(log n) of the optimal size. Based on the approximation algorithm, we design a class of centralized heuristics that are empirically shown to return near-optimal solutions. Simulation results over randomly generated sensor networks with both artificially and naturally generated data sets demonstrate the efficiency of the designed algorithms and the viability of our technique—even in dynamic conditions.

A Real-Time Algorithm for the Approximation of Level-Set-Based Curve Evolution

Abstract: In this paper, we present a complete and practical algorithm for the approximation of level-set-based curve evolution suitable for real-time implementation. In particular, we propose a two-cycle algorithm to approximate level-set-based curve evolution without the need of solving partial differential equations (PDEs). Our algorithm is applicable to a broad class of evolution speeds that can be viewed as composed of a data-dependent term and a curve smoothness regularization term. We achieve curve evolution corresponding to such evolution speeds by separating the evolution process into two different cycles: one cycle for the data-dependent term and a second cycle for the smoothness regularization. The smoothing term is derived from a Gaussian filtering process. In both cycles, the evolution is realized through a simple element switching mechanism between two linked lists, that implicitly represents the curve using an integer valued level-set function. By careful construction, all the key evolution steps require only integer operations. A consequence is that we obtain significant computation speedups compared to exact PDE-based approaches while obtaining excellent agreement with these methods for problems of practical engineering interest. In particular, the resulting algorithm is fast enough for use in real-time video processing applications, which we demonstrate through several image segmentation and video tracking experiments.

Real-time suboptimal model predictive control using a combination of explicit MPC and online optimization

Abstract: Limits on the storage space or the computation time restrict the applicability of model predictive controllers (MPC) in many real problems. Currently available methods either compute the optimal controller online or derive an explicit control law. In this paper we introduce a new approach combining the two paradigms of explicit and online MPC to overcome their individual limitations. The algorithm computes a piecewise affine approximation of the optimal solution that is used to warm-start an active set linear programming procedure. A preprocessing method is introduced that provides hard real-time execution, stability and performance guarantees for the proposed controller. By choosing a combination of the quality of the approximation and the number of online active set iterations the presented procedure offers a tradeoff between the warm-start and online computational effort. We show how the problem of identifying the optimal combination for a given set of requirements on online computation time, storage and performance can be solved. Finally, we demonstrate the potential of the proposed warm-start procedure on numerical examples.

Parameter Estimation in TV Image Restoration Using Variational Distribution Approximation

Abstract: In this paper, we propose novel algorithms for total variation (TV) based image restoration and parameter estimation utilizing variational distribution approximations. Within the hierarchical Bayesian formulation, the reconstructed image and the unknown hyperparameters for the image prior and the noise are simultaneously estimated. The proposed algorithms provide approximations to the posterior distributions of the latent variables using variational methods. We show that some of the current approaches to TV-based image restoration are special cases of our framework. Experimental results show that the proposed approaches provide competitive performance without any assumptions about unknown hyperparameters and clearly outperform existing methods when additional information is included.

Approximation algorithms for clustering uncertain data

Abstract: There is an increasing quantity of data with uncertainty arising from applications such as sensor network measurements, record linkage, and as output of mining algorithms. This uncertainty is typically formalized as probability density functions over tuple values. Beyond storing and processing such data in a DBMS, it is necessary to perform other data analysis tasks such as data mining. We study the core mining problem of clustering on uncertain data, and define appropriate natural generalizations of standard clustering optimization criteria. Two variations arise, depending on whether a point is automatically associated with its optimal center, or whether it must be assigned to a fixed cluster no matter where it is actually located.For uncertain versions of k-means and k-median, we show reductions to their corresponding weighted versions on data with no uncertainties. These are simple in the unassigned case, but require some care for the assigned version. Our most interesting results are for uncertain k-center, which generalizes both traditional k-center and k-median objectives. We show a variety of bicriteria approximation algorithms. One picks O(ke--1log2n) centers and achieves a (1 + e) approximation to the best uncertain k-centers. Another picks 2k centers and achieves a constant factor approximation. Collectively, these results are the first known guaranteed approximation algorithms for the problems of clustering uncertain data.