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Near-Optimal Sensor Placement for Linear Inverse Problems  [PDF]
Juri Ranieri,Amina Chebira,Martin Vetterli
Computer Science , 2013, DOI: 10.1109/TSP.2014.2299518
Abstract: A classic problem is the estimation of a set of parameters from measurements collected by only a few sensors. The number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates. Unfortunately, the selection of the optimal sensor locations is intrinsically combinatorial and the available approximation algorithms are not guaranteed to generate good solutions in all cases of interest. We propose FrameSense, a greedy algorithm for the selection of optimal sensor locations. The core cost function of the algorithm is the frame potential, a scalar property of matrices that measures the orthogonality of its rows. Notably, FrameSense is the first algorithm that is near-optimal in terms of mean square error, meaning that its solution is always guaranteed to be close to the optimal one. Moreover, we show with an extensive set of numerical experiments that FrameSense achieves state-of-the-art performance while having the lowest computational cost, when compared to other greedy methods.
BF-k: a Near-optimal Operator Placement Algorithm for In-network Query Processing  [cached]
Jun-hu Zhang,Feng-jing Shao
Journal of Networks , 2010, DOI: 10.4304/jnw.5.10.1118-1126
Abstract: For a query committed on a node on ad-hoc networks, the data and the relational operations related to the query are distributed widely around the autonomous nodes in the underlying network. Under that condition, in-network query processing is most efficient query evaluation strategy which actually places a tree of relational operators such as filters (joins) and aggregations onto nodes so as to minimize the amount of data transmitted in the network for the query evaluation. Thus the challenge of performing in-network query processing goes to the resolution of the operator placement problem, this paper proposes a kind of decentralized and adaptive operator placement algorithm BF-k, which reduces greatly the time complexity of a near-optimal placement of a large number of relational operators for an in-network query processing. The performance of BF-k is also explored in this paper.
A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs  [PDF]
Jeff Erickson,Anastasios Sidiropoulos
Computer Science , 2013,
Abstract: We present a near-optimal polynomial-time approximation algorithm for the asymmetric traveling salesman problem for graphs of bounded orientable or non-orientable genus. Our algorithm achieves an approximation factor of O(f(g)) on graphs with genus g, where f(n) is the best approximation factor achievable in polynomial time on arbitrary n-vertex graphs. In particular, the O(log(n)/loglog(n))-approximation algorithm for general graphs by Asadpour et al. [SODA 2010] immediately implies an O(log(g)/loglog(g))-approximation algorithm for genus-g graphs. Our result improves the O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi [SODA 2011], which applies only to graphs with orientable genus g; ours is the first approximation algorithm for graphs with bounded non-orientable genus. Moreover, using recent progress on approximating the genus of a graph, our O(log(g) / loglog(g))-approximation can be implemented even without an embedding when the input graph has bounded degree. In contrast, the O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi requires a genus-g embedding as part of the input. Finally, our techniques lead to a O(1)-approximation algorithm for ATSP on graphs of genus g, with running time 2^O(g)*n^O(1).
Near-optimal adjacency labeling scheme for power-law graphs  [PDF]
Casper Petersen,Noy Rotbart,Jakob Grue Simonsen,Christian Wulff-Nilsen
Computer Science , 2015,
Abstract: An adjacency labeling scheme is a method that assigns labels to the vertices of a graph such that adjacency between vertices can be inferred directly from the assigned label, without using a centralized data structure. We devise adjacency labeling schemes for the family of power-law graphs. This family that has been used to model many types of networks, e.g. the Internet AS-level graph. Furthermore, we prove an almost matching lower bound for this family. We also provide an asymptotically near- optimal labeling scheme for sparse graphs. Finally, we validate the efficiency of our labeling scheme by an experimental evaluation using both synthetic data and real-world networks of up to hundreds of thousands of vertices.
Near-optimal Anomaly Detection in Graphs using Lovasz Extended Scan Statistic  [PDF]
James Sharpnack,Akshay Krishnamurthy,Aarti Singh
Statistics , 2013,
Abstract: The detection of anomalous activity in graphs is a statistical problem that arises in many applications, such as network surveillance, disease outbreak detection, and activity monitoring in social networks. Beyond its wide applicability, graph structured anomaly detection serves as a case study in the difficulty of balancing computational complexity with statistical power. In this work, we develop from first principles the generalized likelihood ratio test for determining if there is a well connected region of activation over the vertices in the graph in Gaussian noise. Because this test is computationally infeasible, we provide a relaxation, called the Lovasz extended scan statistic (LESS) that uses submodularity to approximate the intractable generalized likelihood ratio. We demonstrate a connection between LESS and maximum a-posteriori inference in Markov random fields, which provides us with a poly-time algorithm for LESS. Using electrical network theory, we are able to control type 1 error for LESS and prove conditions under which LESS is risk consistent. Finally, we consider specific graph models, the torus, k-nearest neighbor graphs, and epsilon-random graphs. We show that on these graphs our results provide near-optimal performance by matching our results to known lower bounds.
Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time  [PDF]
David Eppstein,Maarten L?ffler,Darren Strash
Computer Science , 2010,
Abstract: The degeneracy of an $n$-vertex graph $G$ is the smallest number $d$ such that every subgraph of $G$ contains a vertex of degree at most $d$. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time $O(dn3^{d/3})$. We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an $n$-vertex graph with degeneracy $d$ (when $d$ is a multiple of 3 and $n\ge d+3$) is $(n-d)3^{d/3}$. Therefore, our algorithm matches the $\Theta(d(n-d)3^{d/3})$ worst-case output size of the problem whenever $n-d=\Omega(n)$.
Near Optimal LP Rounding Algorithm for Correlation Clustering on Complete and Complete k-partite Graphs  [PDF]
Shuchi Chawla,Konstantin Makarychev,Tselil Schramm,Grigory Yaroslavtsev
Computer Science , 2014,
Abstract: We give new rounding schemes for the standard linear programming relaxation of the correlation clustering problem, achieving approximation factors almost matching the integrality gaps: - For complete graphs our appoximation is $2.06 - \varepsilon$ for a fixed constant $\varepsilon$, which almost matches the previously known integrality gap of $2$. - For complete $k$-partite graphs our approximation is $3$. We also show a matching integrality gap. - For complete graphs with edge weights satisfying triangle inequalities and probability constraints, our approximation is $1.5$, and we show an integrality gap of $1.2$. Our results improve a long line of work on approximation algorithms for correlation clustering in complete graphs, previously culminating in a ratio of $2.5$ for the complete case by Ailon, Charikar and Newman (JACM'08). In the weighted complete case satisfying triangle inequalities and probability constraints, the same authors give a $2$-approximation; for the bipartite case, Ailon, Avigdor-Elgrabli, Liberty and van Zuylen give a $4$-approximation (SICOMP'12).
Optimal Sensor Placement for Intruder Detection  [PDF]
Waseem A. Malik,Nuno C. Martins,Ananthram Swami
Mathematics , 2011,
Abstract: We consider the centralized detection of an intruder, whose location is modeled as uniform across a specified set of points, using an optimally placed team of sensors. These sensors make conditionally independent observations. The local detectors at the sensors are also assumed to be identical, with detection probability $(P_{_{D}})$ and false alarm probability $(P_{_{F}})$. We formulate the problem as an N-ary hypothesis testing problem, jointly optimizing the sensor placement and detection policies at the fusion center. We prove that uniform sensor placement is never strictly optimal when the number of sensors $(M)$ equals the number of placement points $(N)$. We prove that for $N_{2} > N_{1} > M$, where $N_{1},N_{2}$ are number of placement points, the framework utilizing $M$ sensors and $N_{1}$ placement points has the same optimal placement structure as the one utilizing $M$ sensors and $N_{2}$ placement points. For $M\leq 5$ and for fixed $P_{_{D}}$, increasing $P_{_{F}}$ leads to optimal placements that are higher in the majorization-based placement scale. Similarly for $M\leq 5$ and for fixed $P_{_{F}}$, increasing $P_{_{D}}$ leads to optimal placements that are higher in the majorization-based placement scale. For $M>5$, this result does not necessarily hold and we provide a simple counterexample. It is conjectured that the set of optimal placements for a given $(M,N)$ can always be placed on a majorization-based placement scale.
Optimal obstacle placement with disambiguations  [PDF]
Vural Aksakalli,Elvan Ceyhan
Statistics , 2012, DOI: 10.1214/12-AOAS556
Abstract: We introduce the optimal obstacle placement with disambiguations problem wherein the goal is to place true obstacles in an environment cluttered with false obstacles so as to maximize the total traversal length of a navigating agent (NAVA). Prior to the traversal, the NAVA is given location information and probabilistic estimates of each disk-shaped hindrance (hereinafter referred to as disk) being a true obstacle. The NAVA can disambiguate a disk's status only when situated on its boundary. There exists an obstacle placing agent (OPA) that locates obstacles prior to the NAVA's traversal. The goal of the OPA is to place true obstacles in between the clutter in such a way that the NAVA's traversal length is maximized in a game-theoretic sense. We assume the OPA knows the clutter spatial distribution type, but not the exact locations of clutter disks. We analyze the traversal length using repeated measures analysis of variance for various obstacle number, obstacle placing scheme and clutter spatial distribution type combinations in order to identify the optimal combination. Our results indicate that as the clutter becomes more regular (clustered), the NAVA's traversal length gets longer (shorter). On the other hand, the traversal length tends to follow a concave-down trend as the number of obstacles increases. We also provide a case study on a real-world maritime minefield data set.
Optimal placement of electrodes in an electroporation process  [cached]
C?ndea Nicolae,Fabrèges Beno?t,de Gournay Frédéric,Poignard Clair
ESAIM : Proceedings , 2010, DOI: 10.1051/proc/2010004
Abstract: Electroporation consists in increasing the permeability of a tissue by applying high voltage pulses. In this paper we discuss the question of optimal placement and optimal loading of electrodes such that electroporation holds only in a given open set of the domain. The electroporated set of the domain is where the norm of the electric field is above a given threshold value. We use a standard gradient algorithm to optimize the loading of the electrodes and shape sensitivity analysis and a gradient algorithm in order to move the electrodes. We also discuss the choice of objective functions to be chosen in the gradient algorithm. L’électroporation est un procédé qui consiste à augmenter la perméabilité d’un tissu en le soumettant à des pulses électriques de grande amplitude. L’objet de ce papier est d’ étudier l’optimisation du placement des électrodes afin d’électroporer une région d’un domaine d’étude. La partie électroporée est celle où la norme du champ électrique dépasse une valeur seuil. Nous utilisons une méthode de gradient standard afin d’optimiser la charge des électrodes. L’optimisation du placement des électrodes est effectuée à l’aide d’une dérivation de domaine et d’un algorithme de gradient. Nous concluons par des simulations qui illustrent l’efficacité de la méthode proposée.
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