Greedy Approximation Research Articles

Gossip algorithms for heterogeneous multi-vehicle routing problems

In this paper we address a class of heterogeneous multi-vehicle task assignment and routing problems. We propose two distributed algorithms based on gossip communication: the first algorithm is based on a local exact optimization and the second is based on a local approximate greedy heuristic. We consider the case where a set of heterogeneous tasks arbitrarily distributed in a plane has to be serviced by a set of mobile robots, each with a given movement speed and task execution speed. Our goal is to minimize the maximum execution time of robots.

Design of viral marketing strategies for product cross-sell through social networks

In this paper, we introduce a novel and generalized version of the influence maximization problem in social networks, which we call as Budgeted Influence Maximization with Cross-sell of Products (B-IMCP), and it considers simultaneously the following three practical aspects: (i) Often cross-sell among products is possible, (ii) Product-specific costs (and benefits) for promoting the products have to be considered, and (iii) Since a company often has budget constraints, the initial seeds have to be chosen within a given budget. In particular, we consider that the cross-sell relationships among the products of a single company are given by an arbitrary bipartite graph. We explore two variants of cross-sell, one weak and one strong, and also assume product-specific costs and benefits. This leads to two different versions of the B-IMCP problem. Given a fixed budget, one of the key issues associated with each version of the B-IMCP problem is to choose the initial seeds within this budget not only for the individual products, but also for promoting cross-sell phenomenon among these products. The following are the specific contributions of this paper: (i) We propose a novel influence propagation model to capture both the cross-sell phenomenon and the costs---benefits for the products; (ii) For each version of the B-IMCP problem, we note that the problem turns out to be NP-hard, and then, we present a simple greedy approximation algorithm for the same. We derive the approximation ratio of this greedy algorithm by drawing upon certain key results from the theory of matroids; (iii) We then outline three heuristics based on well-known concepts from the sociology literature; and (iv) Finally, we experimentally compare and contrast the proposed algorithms and heuristics using certain well-known social network data sets such as WikiVote trust network, Epinions, and Telco call detail records data. Based on the experiments, we consistently found that the stronger the cross-sell relationship between the products, the larger the overlap between the seeds of these products and lesser the distances among the corresponding non-overlapping seeds.

Democratic systems of translates

Democratic systems arise in the context of greedy approximations in Banach spaces. Systems of translates of a single function are the basis of shift invariant subspaces and are used in the construction of wavelets and Gabor systems. In this article, we study the democracy in L2(R) of the system of integer translates of a single function ψ∈L2(R). Necessary and sufficient criteria are given in terms of properties of ψ. The problem of finding an (operative) necessary and sufficient condition is still unsolved.

Lebesgue-Type Inequalities for Greedy Approximation in Banach Spaces

We study sparse representations and sparse approximations with respect to incoherent dictionaries. We address the problem of designing and analyzing greedy methods of approximation. A key question in this regard is: How to measure efficiency of a specific algorithm? Answering this question, we prove the Lebesgue-type inequalities for algorithms under consideration. A very important new ingredient of the paper is that we perform our analysis in a Banach space instead of a Hilbert space. It is known that in many numerical problems, users are satisfied with a Hilbert space setting and do not consider a more general setting in a Banach space. There are known arguments that justify interest in Banach spaces. In this paper, we give one more argument in favor of consideration of greedy approximation in Banach spaces. We introduce a concept of M-coherent dictionary in a Banach space which is a generalization of the corresponding concept in a Hilbert space. We analyze the quasi-orthogonal greedy algorithm (QOGA), which is a generalization of the orthogonal greedy algorithm (orthogonal matching pursuit) for Banach spaces. It is known that the QOGA recovers exactly S -sparse signals after S iterations provided S <; (1+1/M)/2. This result is well known for the orthogonal greedy algorithm in Hilbert spaces. The following question is of great importance: Are there dictionaries in \BBRn such that their coherence in lpn is less than their coherence in l2n for some p ∈ (1,∞)? We show that the answer to the above question is “yes.” Thus, for such dictionaries, replacing the Hilbert space l2n by a Banach space lpn , we improve an upper bound for sparsity that guarantees an exact recovery of a signal.

Approximate Min-Power Strong Connectivity

Given a directed simple graph $G=(V,E)$ and a cost function $c:E \rightarrow R_+$, the power of a vertex $u$ in a directed spanning subgraph $H$ is given by $p_H(u) = \max_{uv \in E(H)} c(uv)$, and corresponds to the energy consumption required for wireless node $u$ to transmit to all nodes $v$ with $uv \in E(H)$. The power of $H$ is given by $p(H) = \sum_{u \in V} p_H(u)$. Power Assignment seeks to minimize $p(H)$ while $H$ satisfies some connectivity constraint. In this paper, we assume $E$ is bidirected (for every directed edge $e \in E$, the opposite edge exists and has the same cost), while $H$ is required to be strongly connected. This is the original power assignment problem introduced by Chen and Huang in 1989, who proved that a bidirected minimum spanning tree has approximation ratio at most 2 (this is tight). In 2010, we introduced a greedy approximation algorithm and claimed a ratio of 1.992. Here we improve the algorithm's analysis to 1.85, combining techniques from Robins and Zelikovsky in 2000 for Steiner Tree, and Caragiannis, Flammini, and Moscardelli in 2007 for the broadcast version of Power Assignment, together with a simple idea inspired by Byrka and coworkers in 2010. The proof also shows that a natural linear programming relaxation, introduced by Calinescu and Qiao in 2012, has integrality gap at most 1.85.

A Novel Sparse Least Squares Support Vector Machines

The solution of a Least Squares Support Vector Machine (LS-SVM) suffers from the problem of nonsparseness. The Forward Least Squares Approximation (FLSA) is a greedy approximation algorithm with a least-squares loss function. This paper proposes a new Support Vector Machine for which the FLSA is the training algorithm—the Forward Least Squares Approximation SVM (FLSA-SVM). A major novelty of this new FLSA-SVM is that the number of support vectors is the regularization parameter for tuning the tradeoff between the generalization ability and the training cost. The FLSA-SVMs can also detect the linear dependencies in vectors of the input Gramian matrix. These attributes together contribute to its extreme sparseness. Experiments on benchmark datasets are presented which show that, compared to various SVM algorithms, the FLSA-SVM is extremely compact, while maintaining a competitive generalization ability.

An improved fast edit approach for two-string approximated mean computation applied to OCR

This paper presents a new fast algorithm for computing an approximation to the mean of two strings of characters representing a 2D shape and its application to a new Wilson-based editing procedure. The approximate mean is built up by including some symbols from the two original strings. In addition, a Greedy approach to this algorithm is studied, which allows us to reduce the time required to compute an approximate mean. The new dataset editing scheme relaxes the criterion for deleting instances proposed by the Wilson editing procedure. In practice, not all instances misclassified by their near neighbors are pruned. Instead, an artificial instance is added to the dataset in the hope of successfully classifying the instance in the future. The new artificial instance is the approximated mean of the misclassified sample and its same-class nearest neighbor.Experiments carried out over three widely known databases of contours show that the proposed algorithm performs very well when computing the mean of two strings, and outperforms methods proposed by other authors. In particular, the low computational time required by the heuristic approach makes it very suitable when dealing with long length strings. Results also show that the proposed preprocessing scheme can reduce the classification error in about 83% of trials. There is empirical evidence that using the Greedy approximation to compute the approximated mean does not affect the performance of the editing procedure.

On greedy approximation algorithms for a class of two-stage stochastic assignment problems

In this paper we consider the two-stage stochastic linear assignment (2SSLA) problem, which is a stochastic extension of the classical deterministic linear assignment problem. For each agent and job, the decision maker has to decide whether to make assignments now or to wait for the second stage. Assignments of agents and jobs, for which decisions are delayed to the second stage, are then completed based on the scenario realized. We discuss two greedy approximation algorithms from the literature and derive a simple necessary optimality condition that generalizes the key ideas behind both of these approaches. Subsequently, based on this result we design a new greedy approximation method. Theoretical observations and the results of computational experiments are also presented.

Keyword-aware optimal route search

Identifying a preferable route is an important problem that finds applications in map services. When a user plans a trip within a city, the user may want to find "a most popular route such that it passes by shopping mall, restaurant , and pub , and the travel time to and from his hotel is within 4 hours." However, none of the algorithms in the existing work on route planning can be used to answer such queries. Motivated by this, we define the problem of keyword-aware optimal route query, denoted by KOR, which is to find an optimal route such that it covers a set of user-specified keywords, a specified budget constraint is satisfied, and an objective score of the route is optimal. The problem of answering KOR queries is NP-hard. We devise an approximation algorithm OSScaling with provable approximation bounds. Based on this algorithm, another more efficient approximation algorithm BucketBound is proposed. We also design a greedy approximation algorithm. Results of empirical studies show that all the proposed algorithms are capable of answering KOR queries efficiently, while the BucketBound and Greedy algorithms run faster. The empirical studies also offer insight into the accuracy of the proposed algorithms.

Reliable Collective Communications With Weighted SRLGs in Optical Networks

In this paper, we study the problem of reliable collective communication (broadcast or gossip) with the objective of maximizing the reliability of the collective communication. The need for collective communication arises in many problems of parallel and distributed computing, including Grid or cloud computing and database management. We describe the network model, formulate the reliable collective communication problem, prove that the maximum reliable collective communication problem is NP-hard, and provide an integer linear program (ILP) formulation for the problem. We then provide a greedy approximation algorithm to construct collective communication (through a spanning tree) that achieves an approximation ratio of 1 + ln(|V| + α|E| - 1), where is the average number of shared link risk groups (SRLGs) along links, and |V| and |E| are the total number of vertices and edges of the network, respectively. Simulations demonstrate that our approximation algorithm achieves good performance in both small and large networks and that, in almost 95% of total cases, our algorithm outperforms the modified minimum spanning tree algorithms.

Analysis of an approximate greedy algorithm for the maximum edge clique partitioning problem

In this note, we show that if the maximum clique problem can be solved by a polynomial time δ-approximation algorithm, then the maximum edge clique partitioning problem (Max-ECP) can be solved by a polynomial time 2(pδ−1)p−1-approximation algorithm for any fixed integer p≥2. This improves the best known bound on the performance ratio of an approximation algorithm for Max-ECP problem and also corrects an error in an earlier work on the topic.

Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators

We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (J. Non-Newtonian Fluid Mech. 139:153-176, 2006) for the numerical solution of high-dimensional Fokker-Planck equations featuring in Navier-Stokes-Fokker-Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson's equation on a rectangular domain in R^2, subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Leli\`evre and Maday (Const. Approx. 30:621-651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173-187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein-Uhlenbeck operator of the kind that appears in Fokker-Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D = D_1 x ... x D_N contained in R^(N d), where each set D_i, i = 1, ..., N, is a bounded open ball in R^d, d = 2, 3.

Certified reduced basis method for electromagnetic scattering and radar cross section estimation

We study nontrivial applications of the reduced basis method (RBM) for electromagnetic applications with an emphasis on scattering and the estimation of radar cross section (RCS). The method and several extensions are explained with two examples with different characteristics. Parameters that are allowed to vary within the model include frequency, incident angle and measurement angle as well as the geometry of the scatterers. With appropriate applications of the empirical interpolation method (EIM), transformation of the domain, configuration of perfectly matched layer, exponential convergence of the reduced basis solution over the entire parameter domain is achieved. Moreover, we demonstrate that this approach allows for the effective capture of the critical behavior, in this case through shapes that minimize scattering. This further highlights the robustness and quality of the greedy approximation and the reduced basis method approach.

The Orthogonal Super Greedy Algorithm and Applications in Compressed Sensing

The general theory of greedy approximation is well developed. Much less is known about how specific features of a dictionary can be used to our advantage. In this paper, we discuss incoherent dictionaries. We build a new greedy algorithm which is called the orthogonal super greedy algorithm (OSGA). We show that the rates of convergence of OSGA and the orthogonal matching pursuit (OMP) with respect to incoherent dictionaries are the same. Based on the analysis of the number of orthogonal projections and the number of iterations, we observed that OSGA is times simpler (more efficient) than OMP. Greedy approximation is also a fundamental tool for sparse signal recovery. The performance of orthogonal multimatching pursuit, a counterpart of OSGA in the compressed sensing setting, is also analyzed under restricted isometry property conditions.

Local Greedy Approximation for Scheduling in Multihop Wireless Networks

In recent years, there has been a significant amount of work done in developing low-complexity scheduling schemes to achieve high performance in multihop wireless networks. A centralized suboptimal scheduling policy, called Greedy Maximal Scheduling (GMS) is a good candidate because its empirically observed performance is close to optimal in a variety of network settings. However, its distributed realization requires high complexity, which becomes a major obstacle for practical implementation. In this paper, we develop simple distributed greedy algorithms for scheduling in multihop wireless networks. We reduce the complexity by relaxing the global ordering requirement of GMS, up to near zero. Simulation results show that the new algorithms approximate the performance of GMS, and outperform the state-of-the-art distributed scheduling policies.

A Lifetime-Preserving and Delay-Constrained Data Gathering Tree for Unreliable Sensor Networks

A tree routing structure is often adopted for many-to-one data gathering and aggregation in sensor networks. For real-time scenarios, considering lossy wireless links, it is an important issue how to construct a maximum-lifetime data gathering tree with delay constraint. In this work, we study the problem of lifetime-preserving and delay-constrained tree construction in unreliable sensor networks. We prove that the problem is NP-complete. A greedy approximation algorithm is proposed. We use expected transmissions count (ETX) as the link quality indicator, as well as a measure of delay. Our algorithm starts from an arbitrary least ETX tree, and iteratively adjusts the hierarchy of the tree to reduce the load on bottleneck nodes by pruning and grafting its sub-tree. The complexity of the proposed algorithm is O(N4). Finally, extensive simulations are carried out to verify our approach. Simulation results show that our algorithm provides longer lifetime in various situations compared to existing data gathering schemes.

New techniques for approximating optimal substructure problems in power-law graphs

The remarkable discovery of many large-scale real networks is the power-law distribution in degree sequence: the number of vertices with degree i is proportional to i−β for some constant β>1. A lot of researchers believe that it may be easier to solve some optimization problems in power-law graphs. Unfortunately, many problems have been proved NP-hard even in power-law graphs. Intuitively, a theoretical question is raised: are these problems on power-law graphs still as hard as on general graphs?In this paper, we show that many optimal substructure problems, such as Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set, are easier to solve in power-law graphs by illustrating better inapproximability factors. An optimization problem has the property of optimal substructure if its optimal solution on some given graph is essentially the union of the optimal sub-solutions on all maximal connected components. In particular, we prove the above problems and a more general problem (ρ-Minimum Dominating Set) remain APX-hard and their constant inapproximability factors on general power-law graphs by using the cycle-based embedding technique to embed any d-bounded graphs into a power-law graph. In addition, in simple power-law graphs, we further prove the corresponding inapproximability factors of these problems based on the graphic embedding technique as well as that of Maximum Clique and Minimum Coloring using the embedding technique in [1]. As a result of these inapproximability factors, the belief that there exists some (1+o(1))-approximation algorithm for these problems on power-law graphs is proven to be not always true. At last, we do in-depth investigations in the relationship between the exponential factor β and constant greedy approximation algorithms.

( k , r )-Dominating set-based, weighted and adaptive clustering algorithms for mobile ad hoc networks

The creation of stable, scalable and adaptive clusters with good performance, faster convergence rate and with minimal overhead is a challenging task in mobile ad hoc networks (MANETs). This study proposes two clustering techniques for MANETs, which are (k, r)-dominating set-based, weighted and adaptive to changes in the network topology. The set of dominating nodes functions as the clusterheads. The scenario-based clustering algorithm for MANETs (SCAM) is a greedy approximation algorithm, whereas the distributed-SCAM (DSCAM) selects the (k, r)-dominating set through a distributed election mechanism. These algorithms achieve variable degree of clusterhead redundancy through the parameter k which contributes to reliability. Similarly, flexibility in creating variable diameter clusters is achieved with the parameter r. To improve the stability of the created clusters, the affiliation of other nodes with the clusterhead is decided based on the quality of the clusterhead, which is a function of connectivity, stability, residual battery power and transmission range. Mechanisms are available for accounting the group mobility and load balancing. The performance of these algorithms are evaluated through simulation and the results show that these algorithms create stable, scalable and load-balanced clusters with relatively less control overhead in comparison with the existing popular algorithms.

A note on ‘Algorithms for connected set cover problem and fault-tolerant connected set cover problem’

A flaw in the greedy approximation algorithm proposed by Zhang et al. (2009) [1] for the minimum connected set cover problem is corrected, and a stronger result on the approximation ratio of the modified greedy algorithm is established. The results are now consistent with the existing results on the connected dominating set problem which is a special case of the minimum connected set cover problem.

Integer programming formulations for the minimum weighted maximal matching problem

Given an undirected graph, the problem of finding a maximal matching that has minimum total weight is NP-hard. This problem has been studied extensively from a graph theoretical point of view. Most of the existing literature considers the problem in some restricted classes of graphs and give polynomial time exact or approximation algorithms. On the contrary, we consider the problem on general graphs and approach it from an optimization point of view. In this paper, we develop integer programming formulations for the minimum weighted maximal matching problem and analyze their efficacy on randomly generated graphs. We also compare solutions found by a greedy approximation algorithm, which is based on the literature, against optimal solutions. Our results show that our integer programming formulations are able to solve medium size instances to optimality and suggest further research for improvement.