Showing papers on "Approximation algorithm published in 2006"

Invitation to fixed-parameter algorithms

Abstract: PART I: FOUNDATIONS 1. Introduction to Fixed-Parameter Algorithms 2. Preliminaries and Agreements 3. Parameterized Complexity Theory - A Primer 4. Vertex Cover - An Illustrative Example 5. The Art of Problem Parameterization 6. Summary and Concluding Remarks PART II: ALGORITHMIC METHODS 7. Data Reduction and Problem Kernels 8. Depth-Bounded Search Trees 9. Dynamic Programming 10. Tree Decompositions of Graphs 11. Further Advanced Techniques 12. Summary and Concluding Remarks PART III: SOME THEORY, SOME CASE STUDIES 13. Parameterized Complexity Theory 14. Connections to Approximation Algorithms 15. Selected Case Studies 16. Zukunftsmusik References Index

Mondrian Multidimensional K-Anonymity

Abstract: K-Anonymity has been proposed as a mechanism for protecting privacy in microdata publishing, and numerous recoding "models" have been considered for achieving 𝑘anonymity This paper proposes a new multidimensional model, which provides an additional degree of flexibility not seen in previous (single-dimensional) approaches Often this flexibility leads to higher-quality anonymizations, as measured both by general-purpose metrics and more specific notions of query answerability Optimal multidimensional anonymization is NP-hard (like previous optimal 𝑘-anonymity problems) However, we introduce a simple greedy approximation algorithm, and experimental results show that this greedy algorithm frequently leads to more desirable anonymizations than exhaustive optimal algorithms for two single-dimensional models

Near-optimal sensor placements: maximizing information while minimizing communication cost

Abstract: When monitoring spatial phenomena with wireless sensor networks, selecting the best sensor placements is a fundamental task. Not only should the sensors be informative, but they should also be able to communicate efficiently. In this paper, we present a data-driven approach that addresses the three central aspects of this problem: measuring the predictive quality of a set of sensor locations (regardless of whether sensors were ever placed at these locations), predicting the communication cost involved with these placements, and designing an algorithm with provable quality guarantees that optimizes the NP-hard tradeoff. Specifically, we use data from a pilot deployment to build non-parametric probabilistic models called Gaussian Processes (GPs) both for the spatial phenomena of interest and for the spatial variability of link qualities, which allows us to estimate predictive power and communication cost of un-sensed locations. Surprisingly, uncertainty in the representation of link qualities plays an important role in estimating communication costs. Using these models, we present a novel, polynomial-time, data-driven algorithm, pSPIEL, which selects Sensor Placements at Informative and cost-Effective Locations. Our approach exploits two important properties of this problem: submodularity, formalizing the intuition that adding a node to a small deployment can help more than adding a node to a large deployment; and locality, under which nodes that are far from each other provide almost independent information. Exploiting these properties, we prove strong approximation guarantees for our pSPlEL approach. We also provide extensive experimental validation of this practical approach on several real-world placement problems, and built a complete system implementation on 46 Tmote Sky motes, demonstrating significant advantages over existing methods.

Balanced Graph Partitioning

Abstract: We consider the problem of partitioning a graph into k components of roughly equal size while minimizing the capacity of the edges between different components of the cut. In particular we require that for a parameter ν ≥ 1, no component contains more than ν · n/k of the graph vertices.For k = 2 and ν = 1 this problem is equivalent to the well-known Minimum Bisection problem for which an approximation algorithm with a polylogarithmic approximation guarantee has been presented in [FK]. For arbitrary k and ν ≥ 2 a bicriteria approximation ratio of O(log n) was obtained by Even et al. [ENRS1] using the spreading metrics technique.We present a bicriteria approximation algorithm that for any constant ν > 1 runs in polynomial time and guarantees an approximation ratio of O(log1.5n) (for a precise statement of the main result see Theorem 6). For ν = 1 and k ≥ 3 we show that no polynomial time approximation algorithm can guarantee a finite approximation ratio unless P = NP.

Connected sensor cover: self-organization of sensor networks for efficient query execution

Abstract: Spatial query execution is an essential functionality of a sensor network, where a query gathers sensor data within a specific geographic region. Redundancy within a sensor network can be exploited to reduce the communication cost incurred in execution of such queries. Any reduction in communication cost would result in an efficient use of the battery energy, which is very limited in sensors. One approach to reduce the communication cost of a query is to self-organize the network, in response to a query, into a topology that involves only a small subset of the sensors sufficient to process the query. The query is then executed using only the sensors in the constructed topology. The self-organization technique is beneficial for queries that run sufficiently long to amortize the communication cost incurred in self-organization. In this paper, we design and analyze algorithms for suchself-organization of a sensor network to reduce energy consumption. In particular, we develop the notion of a connected sensor cover and design a centralized approximation algorithm that constructs a topology involving a near-optimal connected sensor cover. We prove that the size of the constructed topology is within an O(logn) factor of the optimal size, where n is the network size. We develop a distributed self-organization version of the approximation algorithm, and propose several optimizations to reduce the communication overhead of the algorithm. We also design another distributed algorithm based on node priorities that has a further lower communication overhead, but does not provide any guarantee on the size of the connected sensor cover constructed. Finally, we evaluate the distributed algorithms using simulations and show that our approaches results in significant communication cost reductions.

Achieving anonymity via clustering

Abstract: Publishing data for analysis from a table containing personal records, while maintaining individual privacy, is a problem of increasing importance today. The traditional approach of de-identifying records is to remove identifying fields such as social security number, name etc. However, recent research has shown that a large fraction of the US population can be identified using non-key attributes (called quasi-identifiers) such as date of birth, gender, and zip code [15]. Sweeney [16] proposed the k-anonymity model for privacy where non-key attributes that leak information are suppressed or generalized so that, for every record in the modified table, there are at least k−1 other records having exactly the same values for quasi-identifiers. We propose a new method for anonymizing data records, where quasi-identifiers of data records are first clustered and then cluster centers are published. To ensure privacy of the data records, we impose the constraint that each cluster must contain no fewer than a pre-specified number of data records. This technique is more general since we have a much larger choice for cluster centers than k-Anonymity. In many cases, it lets us release a lot more information without compromising privacy. We also provide constant-factor approximation algorithms to come up with such a clustering. This is the first set of algorithms for the anonymization problem where the performance is independent of the anonymity parameter k. We further observe that a few outlier points can significantly increase the cost of anonymization. Hence, we extend our algorithms to allow an e fraction of points to remain unclustered, i.e., deleted from the anonymized publication. Thus, by not releasing a small fraction of the database records, we can ensure that the data published for analysis has less distortion and hence is more useful. Our approximation algorithms for new clustering objectives are of independent interest and could be applicable in other clustering scenarios as well.

Correlation clustering in general weighted graphs

Abstract: We consider the following general correlation-clustering problem [N Bansal, A Blum, S Chawla, Correlation clustering, in: Proc 43rd Annu IEEE Symp on Foundations of Computer Science, Vancouver, Canada, November 2002, pp 238-250]: given a graph with real nonnegative edge weights and a 〈+〉/〈-〉 edge labelling, partition the vertices into clusters to minimize the total weight of cut 〈+〉 edges and uncut 〈-〉 edges Thus, 〈+〉 edges with large weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster while 〈-〉 edges with large weights encourage the endpoints to belong to different clusters In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are effectively chosen to be best possible by the problem definitionCorrelation clustering was introduced by Bansal et al [Correlation clustering, in: Proc 43rd Annu IEEE Syrup on Foundations of Computer Science, Vancouver, Canada, November 2002, pp 238-250], motivated by both document clustering and agnostic learning They proved NP-hardness and gave constant-factor approximation algorithms for the special case in which the graph is complete (full information) and every edge has the same weight We give an O(log n)-approximation algorithm for the general case based on a linear-programming rounding and the "region-growing" technique We also prove that this linear program has a gap of Ω(log n), and therefore our approximation is tight under this approach We also give an O(r3)-approximation algorithm for Kr, r-minor-free graphs On the other hand, we show that the problem is equivalent to minimum multicut, and therefore APX-hard and difficult to approximate better than Θ(log n)

Dependent rounding and its applications to approximation algorithms

Abstract: We develop a new randomized rounding approach for fractional vectors defined on the edge-sets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to improved (approximation) algorithms for various problems. These include:---low congestion multi-path routing;---richer random-graph models for graphs with a given degree-sequence;---improved approximation algorithms for: (i) throughput-maximization in broadcast scheduling, (ii) delay-minimization in broadcast scheduling, as well as (iii) capacitated vertex cover; and---fair scheduling of jobs on unrelated parallel machines.

Computing longest duration flocks in trajectory data

Abstract: Moving point object data can be analyzed through the discovery of patterns. We consider the computational efficiency of computing two of the most basic spatio-temporal patterns in trajectories, namely flocks and meetings. The patterns are large enough subgroups of the moving point objects that exhibit similar movement and proximity for a certain amount of time. We consider the problem of computing a longest duration flock or meeting. We give several exact and approximation algorithms, and also show that some variants are as hard as MaxClique to compute and approximate.

Ruling Out PTAS for Graph Min-Bisection, Dense k-Subgraph, and Bipartite Clique

Abstract: Assuming that NP $ ot\subseteq$ $\cap_{\epsilon > 0}$ BPTIME($2^{n^\epsilon}$), we show that graph min-bisection, dense $k$-subgraph, and bipartite clique have no polynomial time approximation scheme (PTAS). We give a reduction from the minimum distance of code (MDC) problem. Starting with an instance of MDC, we build a quasi-random probabilistically checkable proof (PCP) that suffices to prove the desired inapproximability results. In a quasi-random PCP, the query pattern of the verifier looks random in a certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial, and it can be checked probabilistically by reading a constant number of its values.

Tight approximation algorithms for maximum general assignment problems

Abstract: A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin - ie for bin i, a family Li of subsets of items that fit in bin i The goal is to pack items into bins to maximize the aggregate value This class of problems includes the maximum generalized assignment problem (GAP)1) and a distributed caching problem (DCP) described in this paperGiven a β-approximation algorithm for finding the highest value packing of a single bin, we give1 A polynomial-time LP-rounding based ((1 − 1/e)β)-approximation algorithm2 A simple polynomial-time local search (β/β+1 - e) - approximation algorithm, for any e > 0Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LP-based algorithm with (1 - 1/e - e)-approximation and a local search algorithm with (1/2-e)-approximation guarantee Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 - 1/e The best previously known approximation algorithm for GAP is a 1/2-approximation by Shmoys and Tardos; and Chekuri and Khanna Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gapTo complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 -1/e unless NP⊆ DTIME(nO(log log n)), even if there exists a polynomial-time exact algorithm for the single-bin problemWe extend the (1 - 1/e)-approximation algorithm to a nonseparable assignment problem with applications in maximizing revenue for budget-constrained combinatorial auctions and the AdWords assignment problem We generalize the local search algorithm to yield a 1/2-e approximation algorithm for the k-median problem with hard capacities Finally, we study naturally defined game-theoretic versions of these problems, and show that they have price of anarchy of 2 We also prove the existence of cycles of best response moves, and exponentially long best-response paths to (pure or sink) equilibria

Approximation algorithms for allocation problems: Improving the factor of 1 - 1/e

Abstract: Combinatorial allocation problems require allocating items to players in a way that maximizes the total utility. Two such problems received attention recently, and were addressed using the same linear programming (LP) relaxation. In the Maximum Submodular Welfare (SMW) problem, utility functions of players are submodular, and for this case Dobzinski and Schapira [SODA 2006] showed an approximation ratio of 1 - 1/e. In the Generalized Assignment Problem (GAP) utility functions are linear but players also have capacity constraints. GAP admits a (1 - 1/e)- approximation as well, as shown by Fleischer, Goemans, Mirrokni and Sviridenko [SODA 2006]. In both cases, the approximation ratio was in fact shown for a more general version of the problem, for which improving 1 - 1/e is NPhard. In this paper, we show how to improve the 1 - 1/e approximation ratio, both for SMW and for GAP. A common theme in both improvements is the use of a new and optimal Fair Contention Resolution technique. However, each of the improvements involves a different rounding procedure for the above mentioned LP. In addition, we prove APX-hardness results for SMW (such results were known for GAP). An important feature of our hardness results is that they apply even in very restricted settings, e.g. when every player has nonzero utility only for a constant number of items.

The Effectiveness of Lloyd-Type Methods for the k-Means Problem

Abstract: We investigate variants of Lloyd's heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets. We present variants of Lloyd's heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd's heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyd's method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration.

Defining and computing curve-skeletons with medial geodesic function

Abstract: Many applications in geometric modeling, computer graphics, visualization and computer vision benefit from a reduced representation called curve-skeletons of a shape. These are curves possibly with branches which compactly represent the shape geometry and topology. The lack of a proper mathematical definition has been a bottleneck in developing and applying the the curve-skeletons. A set of desirable properties of these skeletons has been identified and the existing algorithms try to satisfy these properties mainly through a procedural definition. We define a function called medial geodesic on the medial axis which leads to a methematical definition and an approximation algorithm for curve-skeletons. Empirical study shows that the algorithm is robust against noise, operates well with a single user parameter, and produces curve-skeletons with the desirable properties. Moreover, the curve-skeletons can be associated with additional attributes that follow naturally from the definition. These attributes capture shape eccentricity, a local measure of how far a shape is away from a tubular one.

Complex Quadratic Optimization and Semidefinite Programming

Abstract: In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson J. Comput. System Sci., 68 (2004), pp. 442-470]. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex semidefinite programming relaxation problem. In particular, we present an $[m^2(1-\cos\frac{2\pi}{m})/8\pi]$-approximation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of $\pi/4 \approx 0.7854$, which is better than the ratio of $2/\pi \approx 0.6366$ for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with nonpositive off-diagonal elements, then the performance ratio improves to 0.9349.

Network Discovery and Verification

Abstract: Due to its fast, dynamic, and distributed growth process, it is hard to obtain an accurate map of the Internet. In many cases, such a map-representing the structure of the Internet as a graph with nodes and links-is a prerequisite when investigating properties of the Internet. A common way to obtain such maps is to make certain local measurements at a small subset of the nodes, and then to combine these in order to "discover" (an approximation of) the actual graph. Each of these measurements is potentially quite costly. It is thus a natural objective to minimize the number of measurements which still discover the whole graph. We formalize this problem as a combinatorial optimization problem and consider it for two different models characterized by different types of measurements. We give several upper and lower bounds on the competitive ratio (for the online network discovery problem) and the approximation ratio (for the offline network verification problem) in both models. Furthermore, for one of the two models, we compare four simple greedy strategies in an experimental analysis

Are all objectives necessary? on dimensionality reduction in evolutionary multiobjective optimization

Abstract: Most of the available multiobjective evolutionary algorithms (MOEA) for approximating the Pareto set have been designed for and tested on low dimensional problems (≤3 objectives). However, it is known that problems with a high number of objectives cause additional difficulties in terms of the quality of the Pareto set approximation and running time. Furthermore, the decision making process becomes the harder the more objectives are involved. In this context, the question arises whether all objectives are necessary to preserve the problem characteristics. One may also ask under which conditions such an objective reduction is feasible, and how a minimum set of objectives can be computed. In this paper, we propose a general mathematical framework, suited to answer these three questions, and corresponding algorithms, exact and heuristic ones. The heuristic variants are geared towards direct integration into the evolutionary search process. Moreover, extensive experiments for four well-known test problems show that substantial dimensionality reductions are possible on the basis of the proposed methodology.

Max-min surrogate-assisted evolutionary algorithm for robust design

Abstract: Solving design optimization problems using evolutionary algorithms has always been perceived as finding the optimal solution over the entire search space. However, the global optima may not always be the most desirable solution in many real-world engineering design problems. In practice, if the global optimal solution is very sensitive to uncertainties, for example, small changes in design variables or operating conditions, then it may not be appropriate to use this highly sensitive solution. In this paper, we focus on combining evolutionary algorithms with function approximation techniques for robust design. In particular, we investigate the application of robust genetic algorithms to problems with high dimensions. Subsequently, we present a novel evolutionary algorithm based on the combination of a max-min optimization strategy with a Baldwinian trust-region framework employing local surrogate models for reducing the computational cost associated with robust design problems. Empirical results are presented for synthetic test functions and aerodynamic shape design problems to demonstrate that the proposed algorithm converges to robust optimum designs on a limited computational budget

An improved approximation algorithm for combinatorial auctions with submodular bidders

Abstract: We explore the allocation problem in combinatorial auctions with submodular bidders We provide an e/e-1 approximation algorithm for this problem Moreover, our algorithm applies to the more general class of XOS bidders By presenting a matching unconditional lower bound in the communication model, we prove that the upper bound is tight for the XOS classOur algorithm improves upon the previously known 2-approximation algorithm In fact, we also exhibit another algorithm which obtains an approximation ratio better than 2 for submodular bidders, even in the value queries modelThroughout the paper we highlight interesting connections between combinatorial auctions with XOS and submodular bidders and various other combinatorial optimization problems In particular, we discuss coverage problems and online problems

Approximation Algorithms for Metric Facility Location Problems

Abstract: In this paper we present a 1.52-approximation algorithm for the metric uncapacitated facility location problem, and a 2-approximation algorithm for the metric capacitated facility location problem with soft capacities. Both these algorithms improve the best previously known approximation factor for the corresponding problem, and our soft-capacitated facility location algorithm achieves the integrality gap of the standard linear programming relaxation of the problem. Furthermore, we will show, using a result of Thorup, that our algorithms can be implemented in quasi-linear time.

Approximate Bayesian multibody tracking

Abstract: Visual tracking of multiple targets is a challenging problem, especially when efficiency is an issue. Occlusions, if not properly handled, are a major source of failure. Solutions supporting principled occlusion reasoning have been proposed but are yet unpractical for online applications. This paper presents a new solution which effectively manages the trade-off between reliable modeling and computational efficiency. The hybrid joint-separable (HJS) filter is derived from a joint Bayesian formulation of the problem, and shown to be efficient while optimal in terms of compact belief representation. Computational efficiency is achieved by employing a Markov random field approximation to joint dynamics and an incremental algorithm for posterior update with an appearance likelihood that implements a physically-based model of the occlusion process. A particle filter implementation is proposed which achieves accurate tracking during partial occlusions, while in cases of complete occlusion, tracking hypotheses are bound to estimated occlusion volumes. Experiments show that the proposed algorithm is efficient, robust, and able to resolve long-term occlusions between targets with identical appearance

Network correlated data gathering with explicit communication: NP-completeness and algorithms

Abstract: We consider the problem of correlated data gathering by a network with a sink node and a tree-based communication structure, where the goal is to minimize the total transmission cost of transporting the information collected by the nodes, to the sink node. For source coding of correlated data, we consider a joint entropy-based coding model with explicit communication where coding is simple and the transmission structure optimization is difficult. We first formulate the optimization problem definition in the general case and then we study further a network setting where the entropy conditioning at nodes does not depend on the amount of side information, but only on its availability. We prove that even in this simple case, the optimization problem is NP-hard. We propose some efficient, scalable, and distributed heuristic approximation algorithms for solving this problem and show by numerical simulations that the total transmission cost can be significantly improved over direct transmission or the shortest path tree. We also present an approximation algorithm that provides a tree transmission structure with total cost within a constant factor from the optimal.

On the severity of Braess's paradox: designing networks for selfish users is hard

Abstract: We consider a directed network in which every edge possesses a latency function that specifies the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source vertex s to a destination t as quickly as possible. Since the route chosen by one network user affects the congestion experienced by others, we model the problem as a noncooperative game. Assuming that each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to s-t flows in which all flow paths have equal latency.A natural measure for the performance of a network used by selfish agents is the common latency experienced by users in a Nash equilibrium. Braess's Paradox is the counterintuitive but well-known fact that removing edges from a network can improve its performance. Braess's Paradox motivates the following network design problem: given a network, which edges should be removed to obtain the best flow at Nash equilibrium? Equivalently, given a network of edges that can be built, which subnetwork will exhibit the best performance when used selfishly?We give optimal inapproximability results and approximation algorithms for this network design problem. For example, we prove that there is no approximation algorithm for this problem with approximation ratio less than n/2, where n is the number of network vertices, unless P = NP. We further show that this hardness result is the best possible, by exhibiting an (n/2)-approximation algorithm. We also prove tight inapproximability results when additional structure, such as linearity, is imposed on the network latency functions.Moreover, we prove that an optimal approximation algorithm for these problems is the trivial algorithm: given a network of candidate edges, build the entire network. As a consequence, we show that Braess's Paradox--even in its worst-possible manifestations--is impossible to detect efficiently.En route to these results, we give a fundamental generalization of Braess's Paradox: the improvement in performance that can be effected by removing edges can be arbitrarily large in large networks. Even though Braess's Paradox has enjoyed 35 years as a textbook example, our result is the first to extend its severity beyond that in Braess's original four-node network.

Near-optimal algorithms for unique games

Abstract: Unique games are constraint satisfaction problems that can be viewed as a generalization of Max-Cut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none are satisfiable. It has been shown to imply a number of inapproximability results for fundamental problems that seem difficult to obtain by more standard complexity assumptions. Thus, proving or refuting this conjecture is an important goal. We present significantly improved approximation algorithms for unique games. For instances with domain size k where the optimal solution satisfies 1-e fraction of all constraints, our algorithms satisfy roughly k-e/(2-e) and 1- O(√elog k) fraction of all constraints. Our algorithms are based on rounding a natural semidefinite programming relaxation for the problem and their performance almost matches the integrality gap of this relaxation. Our results are near optimal if the Unique Games Conjecture is true, i.e. any improvement (beyond low order terms) would refute the conjecture.

Efficient search for robust solutions by means of evolutionary algorithms and fitness approximation

Abstract: For many real-world optimization problems, the robustness of a solution is of great importance in addition to the solution's quality. By robustness, we mean that small deviations from the original design, e.g., due to manufacturing tolerances, should be tolerated without a severe loss of quality. One way to achieve that goal is to evaluate each solution under a number of different scenarios and use the average solution quality as fitness. However, this approach is often impractical, because the cost for evaluating each individual several times is unacceptable. In this paper, we present a new and efficient approach to estimating a solution's expected quality and variance. We propose to construct local approximate models of the fitness function and then use these approximate models to estimate expected fitness and variance. Based on a variety of test functions, we demonstrate empirically that our approach significantly outperforms the implicit averaging approach, as well as the explicit averaging approaches using existing estimation techniques reported in the literature

New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds

Abstract: There are many applications related to singly linearly constrained quadratic programs subjected to upper and lower bounds. In this paper, a new algorithm based on secant approximation is provided for the case in which the Hessian matrix is diagonal and positive definite. To deal with the general case where the Hessian is not diagonal, a new efficient projected gradient algorithm is proposed. The basic features of the projected gradient algorithm are: 1) a new formula is used for the stepsize; 2) a recently-established adaptive non-monotone line search is incorporated; and 3) the optimal stepsize is determined by quadratic interpolation if the non-monotone line search criterion fails to be satisfied. Numerical experiments on large-scale random test problems and some medium-scale quadratic programs arising in the training of Support Vector Machines demonstrate the usefulness of these algorithms.

Correlation Clustering with a Fixed Number of Clusters

Abstract: We continue the investigation of problems concerning correlation clustering or clustering with qualitative information, which is a clustering formulation that has been studied recently [5, 7, 8, 3] The basic setup here is that we are given as input a complete graph on n nodes (which correspond to nodes to be clustered) whose edges are labeled + (for similar pairs of items) and - (for dissimilar pairs of items) Thus we have only as input qualitative information on similarity and no quantitative distance measure between items The quality of a clustering is measured in terms of its number of agreements, which is simply the number of edges it correctly classifies, that is the sum of number of - edges whose endpoints it places in different clusters plus the number of + edges both of whose endpoints it places within the same clusterIn this paper, we study the problem of finding clusterings that maximize the number of agreements, and the complementary minimization version where we seek clusterings that minimize the number of disagreements We focus on the situation when the number of clusters is stipulated to be a small constant k Our main result is that for every k, there is a polynomial time approximation scheme for both maximizing agreements and minimizing disagreements (The problems are NP-hard for every k ≥ 2) The main technical work is for the minimization version, as the PTAS for maximizing agreements follows along the lines of the property tester for Max k-CUT from [13]In contrast, when the number of clusters is not specified, the problem of minimizing disagreements was shown to be APX-hard [7], even though the maximization version admits a PTAS