Performance Of Greedy Algorithm Research Articles

A new hybrid genetic algorithm with tabu search for solving the temporal coverage problem using rotating directional sensors

AbstractOne of the most important problems in directional sensor networks is coverage problem. The coverage can be measured in two ways: positional or temporal. In temporal coverage, the directional sensors rotate periodically round themselves in a repetitive process. Thus, in each time slot, those targets that are positioned within the sensor nodes radius receive their desired coverage. In this model, if a target is left uncovered, it is said that the target has remained in darkness. The main task defined for the temporal coverage model is the minimization of the total dark time for all the targets in the network. This problem has been solved by greedy‐based algorithms in last studies. Greedy‐based algorithms are able to solve the temporal coverage problem in real time. Remember that the performance of greedy algorithms is extremely dependent on the closeness of optimal solution and initial candidates. For this reason, greedy algorithms may obtain local minima due to heuristic search. As far as we know meta‐heuristic algorithms have not been used in past researches to solve such problems. For solving this problem, in this paper two algorithms were developed, GA‐based and hybridized model comprising genetic algorithms and tabu search. A new model was suggested for the chromosome in genetic algorithm. To evaluate the performance of the developed algorithms, they were compared with randomized scenario and greedy‐based algorithm presented in last studies. For better comparison, several parameters, including total dark time, number of sensors, number of targets, sector angle, sensing range were taken into account. The results obtained from the comparison of the algorithms indicated that the developed algorithms are effective in solving the temporal coverage problem in terms of minimizing the total dark time of the targets.

Performance of community detection algorithms supported by node embeddings

Abstract The grouping of nodes into subsets that are relatively densely interconnected and separable from the rest of the network is a property often displayed in many complex real-world networks; this feature is known as a community structure. There is a growing demand for algorithms that can find partitions that resemble the community structure of a given network as closely as possible. However, most popular algorithms for community detection in graphs have one serious drawback, namely, they are heuristic-based and in many cases are unable to find a near-optimal solution. Moreover, their results are volatile, impacting the replicability of their results. In this paper, we investigate if the performance of greedy algorithms might be improved by initialising such algorithms with some carefully chosen partition of nodes, namely a partition obtained by embedding the nodes into real numbers space and then running a clustering algorithm on this latent representation. We believe that embedding will filter unwanted noise while retaining the proximity of nodes belonging to the same community or will learn more complex and elusive relations between nodes. Then, clustering algorithms run on this embedding will create a stable partitioning that will reduce the uncertainty in the initial phases of the community detection algorithms. The experiments show that the proposed procedure significantly improves the results over baseline community detection algorithms, namely Louvain and Leiden. It also reduces the inherent volatility of such algorithms. The impact depends on the given graph’s properties, especially the strength of the community structure and degree distribution. The largest boost in performance is given in the cases when networks are ‘noisier’, that is, when the community structure is less pronounced and there are many connections between communities. Furthermore, the design and parametrization of the procedure depend on the network’s topology, not on the size of the network itself.

Influence Maximization on Undirected Graphs

We study the influence maximization problem in undirected networks, specifically focusing on the independent cascade and linear threshold models. We prove APX-hardness (NP-hardness of approximation within factor (1-τ) for some constant τ > 0) for both models, which improves the previous NP-hardness lower bound for the linear threshold model. No previous hardness result was known for the independent cascade model. As part of the hardness proof, we show some natural properties of these cascades on undirected graphs. For example, we show that the expected number of infections of a seed set S is upper bounded by the size of the edge cut of S in the linear threshold model and a special case of the independent cascade model, the weighted independent cascade model. Motivated by our upper bounds, we present a suite of highly scalable local greedy heuristics for the influence maximization problem on both the linear threshold model and the weighted independent cascade model on undirected graphs that, in practice, find seed sets that on average obtain 97.52% of the performance of the much slower greedy algorithm for the linear threshold model and 97.39% of the performance of the greedy algorithm for the weighted independent cascade model. Our heuristics also outperform other popular local heuristics, such as the degree discount heuristic by Chen et al. [7].

Greedy algorithms and Kolmogorov widths in Banach spaces

Let X be a Banach space and K be a compact subset in X. We consider a greedy algorithm for finding n-dimensional subspace Vn⊂X which can be used to approximate the elements of K. We are interested in how well the space Vn approximates the elements of K. For this purpose we compare the performance of greedy algorithm measured by σn(K)X≔dist(K,Vn)X with the Kolmogorov width dn(K)X which is the best possible error one can achieve when approximating K by n-dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (2011), DeVore et al. (2013) and Wojtaszczyk (2015). The purpose of the present paper is to continue this line. We shall show that there exists a constant C>0 such that σn(K)X≤Cn−s+μ(log(n+1))min(s,1∕2),n≥1, if Kolmogorov widths dn(K)X decay as n−s and the Banach–Mazur distance between an arbitrary n-dimensional subspace Vn⊂X and ℓ2n satisfies d(Vn,ℓ2n)≤C1nμ. In particular, when some additional information about the set K is given then there is no logarithmic factor in this estimate.

Edge guided compressive sensing for image reconstruction based on two-stage [formula omitted] minimization

In compressive sensing framework, the results of image reconstruction are sometimes not accurate enough due to the downsampled measurements, especially when the sampling rate is relatively small. This paper proposes a novel edge guided compressive sensing (EGCS) algorithm for natural image reconstruction based on two-stage l0 minimization, aiming to improve the reconstruction performance. Firstly, wavelet transform is utilized to provide sparsity and multiple sampling scheme is employed to acquire the down-sampled measurements. Then, in the first stage, we design an edge-preserving smoothing method by l0 gradient minimization to extract the important edge prior accurately, which can not only contribute a lot to improve the reconstruction accuracy of image structures but also reduce the computational complexity remarkably. Also, the use of multiple sampling scheme is beneficial to enhancing the guidance accuracy of multiple edge prior. In the second stage, under the guidance of multiple edge prior, the intelligent searching strategies are designed by taking advantages of intelligent optimization algorithms in solving combinatorial optimization problems and utilizing the superior performance of greedy algorithm on reconstruction speed, which is conductive to solve the joint sparse reconstruction based on l0 minimization essentially. The better reconstruction performance can be achieved based on the fact that it is more likely to find the global optimal solution accurately by the designed intelligent searching strategies, especially when the sampling rate is relatively small. Experimental results on natural image reconstruction demonstrate that our proposed method EGCS is superior to the state-of-the-art reconstruction algorithms, and can well preserve the important image structures at the same time.

Distributed Submodular Maximization With Limited Information

We consider a class of distributed submodular maximization problems in which each agent must choose a single strategy from its strategy set. The global objective is to maximize a submodular function of the strategies chosen by each agent. When choosing a strategy, each agent has access to only a limited number of other agents’ choices. For each of its strategies, an agent can evaluate its marginal contribution to the global objective given its information. The main objective is to investigate how this limitation of information about the strategies chosen by other agents affects the performance when agents make choices according to a local greedy algorithm. In particular, we provide lower bounds on the performance of greedy algorithms for submodular maximization, which depend on the clique number of a graph that captures the information structure. We also characterize graph-theoretic upper bounds in terms of the chromatic number of the graph. Finally, we demonstrate how certain graph properties limit the performance of the greedy algorithm. Simulations on several common models for random networks demonstrate our results.

Comparative analysis on the selection of number of clusters in community detection.

We conduct a comparative analysis on various estimates of the number of clusters in community detection. An exhaustive comparison requires testing of all possible combinations of frameworks, algorithms, and assessment criteria. In this paper we focus on the framework based on a stochastic block model, and investigate the performance of greedy algorithms, statistical inference, and spectral methods. For the assessment criteria, we consider modularity, map equation, Bethe free energy, prediction errors, and isolated eigenvalues. From the analysis, the tendency of overfit and underfit that the assessment criteria and algorithms have becomes apparent. In addition, we propose that the alluvial diagram is a suitable tool to visualize statistical inference results and can be useful to determine the number of clusters.

Sensor selection for Kalman filtering of linear dynamical systems: Complexity, limitations and greedy algorithms

We consider the problem of selecting an optimal set of sensors to estimate the states of linear dynamical systems. Specifically, the goal is to choose (at design-time) a subset of sensors (satisfying certain budget constraints) from a given set in order to minimize the trace of the steady state a priori or a posteriori error covariance produced by a Kalman filter. We show that the a priori and a posteriori error covariance-based sensor selection problems are both NP-hard, even under the additional assumption that the system is stable. We then provide bounds on the worst-case performance of sensor selection algorithms based on the system dynamics, and show that greedy algorithms are optimal for a certain class of systems. However, as a negative result, we show that certain typical objective functions are not submodular or supermodular in general. While this makes it difficult to evaluate the performance of greedy algorithms for sensor selection (outside of certain special cases), we show via simulations that these greedy algorithms perform well in practice.

Shrinkage-Based Alternating Projection Algorithm for Efficient Measurement Matrix Construction in Compressive Sensing

A simple but efficient measurement matrix construction algorithm (MMCA) within compressive sensing (CS) framework is introduced. In the CS framework, the smaller coherence between the measurement matrix Φ and the sparse matrix (basis) Ψ can lead to better signal reconstruction performance. In this paper, we achieve this purpose by adopting shrinkage and alternating projection technique iteratively. Finally, the coherence among the columns of the optimized measurement matrix Φ and the fixed sparse matrix Ψ can be decreased greatly, even close to the Welch bound. The extensive experiments have been conducted to test the performance of the proposed algorithm, which are compared with that of the state-of-the-art algorithms. We conclude that the recovery performance of greedy algorithms [e.g., orthogonal matching pursuit (OMP) and regularized OMP] using the proposed MMCA outperforms the random algorithm and the algorithms introduced by Elad, Vahid, Hang, and Xu. In addition, the real temperature data gathering and reconstruction in wireless sensor networks have been conducted. The experimental results also show the superiority of MMCA for real temperature data reconstruction comparing with other existing measurement matrix optimization algorithms.

Greedy Algorithms for Reduced Bases in Banach Spaces

Given a Banach space X and one of its compact sets $\mathcal{F}$ , we consider the problem of finding a good n-dimensional space X n ⊂X which can be used to approximate the elements of $\mathcal{F}$ . The best possible error we can achieve for such an approximation is given by the Kolmogorov width $d_{n}(\mathcal{F})_{X}$ . However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) in the case $X=\mathcal{H}$ is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457–1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces.

On the optimality of the Orthogonal Greedy Algorithm for [formula omitted]-coherent dictionaries

In this article, we continue to study the performance of Greedy Algorithms. We show that the Orthogonal Greedy Algorithm (Orthogonal Matching Pursuit) provides an almost optimal approximation on the first [μ−1/20] steps for μ-coherent dictionaries.

On performance of greedy algorithms

We show that the Orthogonal Greedy Algorithm (OGA) for dictionaries in a Hilbert space with small coherence M performs almost as well as the best m -term approximation for all signals with sparsity close to the best theoretically possible threshold m = 1 2 ( M − 1 + 1 ) by proving a Lebesgue-type inequality for arbitrary signals. Additionally, we present a dictionary with coherence M and a 1 2 ( M − 1 + 1 ) -sparse signal for which OGA fails to pick up any atoms from the support, showing that the above threshold is sharp. We also show that the Pure Greedy Algorithm (PGA) matches the rate of convergence of the best m -term approximation beyond the saturation limit of m − 1 2 .

Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections

We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant-degree regular graphs. We show that forr-regular graphs withnnodes and girth at leastg, the algorithm finds an independent set of expected cardinalitywheref(r) is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degreerand girthgare large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm inarbitrarybounded-degree graphs is concentrated around the mean. Finally, we analyse the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.

Approximation and learning by greedy algorithms

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.

On the Performance of Greedy Algorithms in Packet Buffering

We study a basic buffer management problem that arises in network switches. Consider $m$ input ports, each of which is equipped with a buffer (queue) of limited capacity. Data packets arrive online and can be stored in the buffers if space permits; otherwise packet loss occurs. In each time step the switch can transmit one packet from one of the buffers to the output port. The goal is to maximize the number of transmitted packets. Simple arguments show that any work-conserving algorithm, which serves any nonempty buffer, is 2-competitive. Azar and Richter recently presented a randomized online algorithm and gave lower bounds for deterministic and randomized strategies. In practice, greedy algorithms are very important because they are fast, use little extra memory, and reduce packet loss by always serving a longest queue. In this paper we first settle the competitive performance of the entire family of greedy strategies. We prove that greedy algorithms are not better than 2-competitive no matter how ties are broken. Our lower bound proof uses a new recursive construction for building adversarial buffer configurations that may be of independent interest. We also give improved lower bounds for deterministic and randomized online algorithms. In this paper we present the first deterministic online algorithm that is better than 2-competitive. We develop a modified greedy algorithm, called semigreedy, and prove that it achieves a competitive ratio of $17/9 \approx 1.89$. The new algorithm is simple, fast, and uses little extra memory. Only when the risk of packet loss is low does it not serve the longest queue. Additionally we study scenarios when an online algorithm is granted additional resources. We consider resource augmentation with respect to memory and speed; i.e., an online algorithm may be given larger buffers or higher transmission rates. We analyze greedy and other online strategies.

SIGACT news complexity theory column 35

I was surprised and delighted at a recent conference when the complexity class Θ 2 p suddenly popped up in a talk (by Markus Holzer on work by him and Pierre McKenzie) where it is was not exactly something one would have before-hand expected to come into play. After hearing the rest of Markus's remarkable talk (which he illustrated using toys from his daughter), I was much hoping that the authors would be so kind as to write a guest column; they very generously agreed to, and that column appears below.So, what is Θ 2 p ? Θ 2 p is the class of languages acceptable via parallel access to NP, and has many equivalent alternate definitions and characterizations (see [KSW87, Hem89, Wag90]), such as the one used in the paper below. This class was first studied by Papadimitriou and and Zachos in 1983 [PZ83] (though in a form that at that time was not known to be equivalent to the definition just mentioned), and has played an important role in complexity theory, e.g., Kadin [Kad89] proved that the polynomial-time hierarchy collapses to Θ 2 p if NP has sparse Turing-complete sets. Recently, the class has turned out to play other surprising roles, such as in analyzing the complexity of electoral systems (see the survey [HH00]) and in understanding the performance of greedy algorithms [HR98, HRS01]. In fact, a survey column a few years ago was devoted to overviewing such raised-to-Θ 2 p lower bounds [HHR97]. Bill Gasarch and SIGACT News need YOU! That is, the next issue will contain a guest column edited by Bill Gasarch, which will collection opinions on the future of the P-versus-NP question. Bill very much wants a broad range of opinions, and will be setting up a web page (http://www.cs.umd.edu/~gasarch/pnp/) describing how you can get your opinions to him. He may even put up a series of polling questions there, so please go there, vote early, and vote often!And in the issues following that will be articles by M. Schaeffer and C. Umans on complete sets at and beyond the polynomial hierarchy's second level, by A. Nickelsen and T. Tantau on partial information classes, and by R. Paturi on the complexity of k -SAT.

The Performance of greedy algorithms for the on-line steiner tree and related problems

We study the on-line Steiner tree problem on a general metric space. We show that the greedy on-line algorithm isO(log((d/z)s))-competitive, wheres is the number of regular nodes,d is the maximum metric distance between any two revealed nodes, andz is the optimal off-line cost. Our results refine the previous known bound [9] and show that AlgorithmSB of Bartalet al. [3] for the on-line file allocation problem isO(log logN)-competitive on anN-node hypercube or butterfly network. A lower bound of Ω(log((d/z)s)) is shown to hold.