Approximating minimum cost source location problems with local vertex-connectivity demands

We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most m-k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most m-k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size mathcal {O}(k^4t^5) and can be solved in randomized (4t)^kcdot n^{mathcal {O}(1)} time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in k^{2k}cdot n^{mathcal {O}(1)} time on directed acyclic graphs, (iii) Directed Additive Spanner is {{,mathrm{mathsf{W}},}}[1]-hard when parameterized by k for every fixed tge 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is {{,mathrm{mathsf{FPT}},}} when parameterized by t and k.

An approximation algorithm for the Generalized [formula omitted]-Multicut problem

We consider the Maximum Node Disjoint Paths (MNDP) problem in undirected graphs. The input consists of an undirected graph G = (V, E) and a collection {(s1, t1), …, (sk, tk)} of k source-sink pairs. The goal is to select a maximum cardinality subset of pairs that can be routed/connected via node-disjoint paths. A relaxed version of MNDP allows up to c paths to use a node, where c is the congestion parameter. We give a polynomial time algorithm that routes Ω(OPT/poly log k) pairs with O(1) congestion, where OPT is the value of an optimum fractional solution to a natural multicommodity flow relaxation. Our result builds on the recent breakthrough of Chuzhoy [17] who gave the first poly-logarithmic approximation with constant congestion for the Maximum Edge Disjoint Paths (MEDP) problem.

On convexification of range measurement based sensor and source localization problems

A wide range of research has been published on the problem of estimating the parameters of electromagnetic and acoustical sources from measurements of signals measured at an array of sensors. In the quasi-static electromagnetic cases examined here, the signal variation from a point source is relatively slow with respect to the signal propagation and the spacing of the array of sensors. As such, the location of the point sources can only be determined from the spatial diversity of the received signal across the array. The inverse source localization problem is complicated by unknown model order and strong local minima. The nonlinear optimization problem is posed for solving for the parameters of the quasi-static source model. The transient nature of the sources can be exploited to allow subspace approaches to separate out the signal portion of the spatial correlation matrix. Decomposition techniques are examined for improved processing, and an adaptation of MUtiple SIgnal Characterization (MUSIC) is presented for solving the source localization problem. Recent results on calculating the Cramer-Rao error lower bounds are extended to the multidimensional problem here. This thesis focuses on the problem of source localization in magnetoencephalography (MEG), with a secondary application to thunderstorm source localization. Comparisons are also made between MEG and its electrical equivalent, electroencephalography (EEG). The error lower bounds are examined in detail for several MEG and EEG configurations, as well as localizing thunderstorm cells over Cape Canaveral and Kennedy Space Center. Time-eigenspectrum is introduced as a parsing technique for improving the performance of the optimization problem.

A fundamental problem in computer and communication networks is the wavelength assignment (WA) problem: given a set of routing paths on a network, assign wavelengths (channels) to the paths such that the paths with the same wavelength are edge-disjoint. The optimization problem here is to minimize the number of wavelengths. A popular network topology is a tree of rings. It is known NP-hard to find the minimum number of wavelengths for the WA problem on a tree of rings. Let L be the maximum number of paths on any edge in the network. Then L is a lower bound on the number of wavelengths for the WA problem. We give a polynomial time algorithm which uses at most 3L wavelengths for the WA problem on a tree of rings with node degree at most eight. This improves the previous result of 4L. We also show that some instances of the WA problem require at least 3L wavelengths on a tree of rings, implying that the 3L upper bound is optimal for the worst case instances. In addition, we prove that our algorithm has approximation ratios 2 and 2.5 for a tree of rings with node degrees at most four and six, respectively.

This paper deals with the problem of odor source localization by designing and analyzing a decision-control system (DCS) for a group of robots. In the decision level, concentration magnitude information and wind information detected by robots are used to predict a probable position of the odor source. Specifically, the idea of particle swarm optimization is introduced to give a probable position of the odor source in terms of concentration magnitude information. Moreover, an observation model of the position of the odor source is built according to wind information, and a Kalman filter is used to estimate the position of the odor source, which is combined with the position obtained by using concentration magnitude information in order to make a decision on the position of the odor source. In the control level, two types of the finite-time motion control algorithms are designed; one is a finite-time parallel motion control algorithm, while the other is a finite-time circular motion control algorithm. Precisely, a nonlinear finite-time consensus algorithm is first proposed, and a Lyapunov approach is used to analyze the finite-time convergence of the proposed consensus algorithm. Then, on the basis of the proposed finite-time consensus algorithm, a finite-time parallel motion control algorithm, which can control the group of robots to trace the plume and move toward the probable position of odor source, is derived. Next, a finite-time circular motion control algorithm, which can enable the robot group to circle the probable position of the odor source in order to search for odor clues, is also developed. Finally, the performance capabilities of the proposed DCS are illustrated through the problem of odor source localization.

On Designing a Cooperative Control System for Multiple Mobile Robots to Locate the Source of Odor

The problem of source localization from arrival time delay estimates requires a computationally costly iterative solution of a set of nonlinear equations. Most known methods assume that the propagation speed is known. In this paper, we provide several effective source localization and propagation velocity estimation methods which only use measurements of the relative arrival time delays between sensors. The formulae for source localization and propagation speed estimation are derived based on least squares, total least squares, bounded data uncertainty, and constrained least squares methods. Statistical performance of these methods are compared via computer simulation. In addition, in order to avoid time delay ambiguity problems and obtain smoother time delays, two time delay smoothing methods based on the forward backward algorithm and the Viterbi algorithm are also proposed. Field experiment results based on these techniques are also presented.

In this letter, the problem of source localization using time difference of arrival is investigated. Then, a closed form two-stage solution is proposed based on estimation of the range nuisance parameter in the first stage and refinement of initial solution in the next stage. The proposed solution is shown analytically and verified by simulations to be an efficient estimate, which can attain the CRLB performance under mild Gaussian noise assumption. This method is able to locate the source with the minimum number of sensors required for N-dimensional localization. Numerical simulations demonstrate significant performance improvement of the proposed method compared with the state-of-the-art algorithms and corroborate the theoretical developments.

Source location privacy is becoming an increasingly important property in wireless sensor network applications, such as asset monitoring. The original source location problem is to protect the location of a source in a wireless sensor network from a single distributed eavesdropper attack. Several techniques have been proposed to address the source location problem, where most of these apply some form of traffic analysis and engineering to provide enhanced privacy. One such technique, namely fake sources, has proved to be promising for providing source location privacy. Recent research has concentrated on investigating the efficiency of fake source approaches under various attacker models. In this paper, we (i) provide a novel formalisation of the source location privacy problem, (ii) prove the source location privacy problem to be NP-complete, and (iii) provide a heuristic that yields an optimal level of privacy under appropriate parameterisation. Crucially, the results presented show that fake sources can provide a high, sometimes optimal, level of privacy.

We consider the source and network localization problems, seeking to strengthen the relationship between the Weighted-Least-Square (WLS) and the Maximum-Likelihood (ML) solutions of these problems. To this end, we design an optimization algorithm for source and network localization under the principle that: a) the WLS and the ML objectives should be the same; and b) the solution of the ML-WLS objective does not depend on any information besides the set of given distance measurements (observations). The proposed Range-Global Distance Continuation (R-GDC) algorithm solves the localization problems via iterative minimizations of smoothed variations of the WLS objective, each obtained by convolution with a Gaussian kernel of progressively smaller variances. Since the last (not smoothed) WLS objective derives directly from the ML formulation of the localization problem, and the R-GDC requires no initial estimate to minimize it, final result is maximum-likelihood approach to source and network localization problems. The performance of the R-GDC method is compared to that of state-of-the-art techniques such as semidefinite programming (SDP), nonlinear Newton least squares (NLS), and the Stress-of-a-MAjorizing-Complex-Objective-Function (SMACOF) algorithms, as well as to the Cramér-Rao Lower Bound (CRLB). The comparison reveals that the solutions obtained with the R-GDC algorithm is insensitive to initial estimates and provides a localization error that closely approaches that of the corresponding fundamental bounds. The R-GDC is also found to achieve a complexity order comparable to that of the SMACOF, which is known for its efficiency.

A fundamental problem in communication networks is wavelength assignment (WA): given a set of routing paths on a network, assign a wavelength to each path such that the paths with the same wavelength are edge-disjoint, using the minimum number of wavelengths. The WA problem is NP-hard for a tree of rings network which is well used in practice. In this paper, we give an efficient algorithm, which solves the WA problem on a tree of rings with an arbitrary (node) degree by at most 3L wavelengths and achieves an approximation ratio of 2.75 asymptotically, where L is the maximum number of paths on any link in the network. The 3L upper bound is tight that there are instances of the WA problem that require at least 3L wavelengths even on a tree of rings with degree four. We also give a 3L and 2-approximation algorithm for the WA problem on a tree of rings with degree at most six. Previous results include: 4L (resp. 3L) wavelengths for trees of rings with arbitrary degrees (resp. degree at most eight), and 2-approximation (resp. 2.5-approximation) algorithm for trees of rings with degree four (resp. six).

The single-source localization problem (SSLP), which is nonconvex by its nature, appears in several important multidisciplinary fields such as signal processing and the global positioning system. In this paper, we cast SSLP as a Euclidean distance embedding problem and study a Lagrangian dual approach. It is proved that the Lagrangian dual problem must have an optimal solution under the generalized Slater condition. We provide a sufficient condition for the zero-duality gap and establish the equivalence between the Lagrangian dual approach and the existing Generalized Trust-Region Subproblem (GTRS) approach studied by Beck et al. [“Exact and Approximate Solutions of Source Localization Problems,” IEEE Trans. Signal Process., vol. 56, pp. 1770-1778, 2008]. We also reveal new implications of the assumptions made by the GTRS approach. Moreover, the Lagrangian dual approach has a straightforward extension to the multiple-source localization problem. Numerical simulations demonstrate that the Lagrangian dual approach can produce localization of similar quality as the GTRS and can significantly outperform the well-known semidefinite programming solver for the multiple source localization problem on the tested cases.

To locate source nodes of influence propagation given an observation set in online social networks is the source location problem. It is increasingly critical to control the diffusion of malicious information in terms of source nodes with the popularity of online social networks. In this paper, we focus on multi-source location with incomplete information. First, to track the causes that make node activation, we propose the Bayes backtracking model (BBM), where the posterior probabilities of activation causes are formulated. Then, we employ random walk to represent the source backtracking process and present the corresponding backtracking algorithm for source location by taking the posterior probability in BBM as the transition probability in random walk. Finally, we test the effectiveness of our method on real online social networks and make performance studies.