Maximizing the Number of Spanning Trees in a Connected Graph

Li, Huan; Patterson, Stacy; Yi, Yuhao; Zhang, Zhongzhi

doi:10.1109/tit.2019.2940263

Cited by 23 publications

(11 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Improving the computational complexity of the greedy algorithm is another avenue for future work. Our approach has been recently adopted by Li et al (2018) who proposed a faster greedy algorithm with an approximation ratio of ( 1 − 1 / e − ϵ ) for any positive ϵ . Finally, we are ultimately interested in a seamless incorporation of graph topology into planning and decision-making pipelines beyond measurement selection.…”

Section: Discussionmentioning

confidence: 99%

Reliable Graphs for SLAM

Khosoussi

Giamou

Sukhatme

et al. 2019

The International Journal of Robotics Research

View full text Add to dashboard Cite

Estimation-over-graphs (EoG) is a class of estimation problems that admit a natural graphical representation. Several key problems in robotics and sensor networks, including sensor network localization, synchronization over a group, and simultaneous localization and mapping (SLAM) fall into this category. We pursue two main goals in this work. First, we aim to characterize the impact of the graphical structure of SLAM and related problems on estimation reliability. We draw connections between several notions of graph connectivity and various properties of the underlying estimation problem. In particular, we establish results on the impact of the weighted number of spanning trees on the D-optimality criterion in 2D SLAM. These results enable agents to evaluate estimation reliability based only on the graphical representation of the EoG problem. We then use our findings and study the problem of designing sparse SLAM problems that lead to reliable maximum likelihood estimates through the synthesis of sparse graphs with the maximum weighted tree connectivity. Characterizing graphs with the maximum number of spanning trees is an open problem in general. To tackle this problem, we establish several new theoretical results, including the monotone log-submodularity of the weighted number of spanning trees. We exploit these structures and design a complementary greedy–convex pair of efficient approximation algorithms with provable guarantees. The proposed synthesis framework is applied to various forms of the measurement selection problem in resource-constrained SLAM. Our algorithms and theoretical findings are validated using random graphs, existing and new synthetic SLAM benchmarks, and publicly available real pose-graph SLAM datasets.

show abstract

Section: Discussionmentioning

confidence: 99%

Reliable Graphs for SLAM

Khosoussi

Giamou

Sukhatme

et al. 2019

The International Journal of Robotics Research

View full text Add to dashboard Cite

show abstract

“…Admittedly, as a practical approach of graph edit, edge addition operation has been extensively used for different application purposes, such as improving the centrality of a node [12,13,37] and maximizing the number of spanning trees [26]. For a social network, creating edges corresponds to making friends.…”

Section: Related Workmentioning

confidence: 99%

Maximizing Influence of Leaders in Social Networks

Zhou,

Zhang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The operation of adding edges has been frequently used to the study of opinion dynamics in social networks for various purposes. In this paper, we consider the edge addition problem for the DeGroot model of opinion dynamics in a social network with nodes and edges, in the presence of a small number ≪ of competing leaders with binary opposing opinions 0 or 1. Concretely, we pose and investigate the problem of maximizing the equilibrium overall opinion by creating new edges in a candidate edge set, where each edge is incident to a 1-valued leader and a follower node. We show that the objective function is monotone and submodular. We then propose a simple greedy algorithm with an approximation factor (1 − 1 ) that approximately solves the problem in ( 3 ) time. Moreover, we provide a fast algorithm with a (1 − 1 − ) approximation ratio and ˜ ( −2 ) time complexity for any > 0, where ˜ (•) notation suppresses the poly(log ) factors. Extensive experiments demonstrate that our second approximate algorithm is efficient and effective, which scales to large networks with more than a million nodes.

show abstract

“…dynamics [2], [3], [4] and bond percolation [5] on a graph. Concerning the Laplacian matrix, its smallest and largest nonzero eigenvalues are closely related to the time of convergence and delay robustness of the consensus problem [6]; all the nonzero eigenvalues determine the number of spanning trees [7] and the sum of resistance distances over all node pairs [8], [9], with the latter encoding the performance of different dynamical processes, such as the total hitting times of random walks [10], [11], [12] and robustness to noise in consensus problem [13], [14], [15].…”

Section: Introductionmentioning

confidence: 99%

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Wang¹,

Zhou²,

Li³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

For a connected graph G = (V, E) with n nodes, m edges, and Laplacian matrix L, a grounded Laplacian matrix L(S) of G is a (n − k) × (n − k) principal submatrix of L, obtained from L by deleting k rows and columns corresponding to k selected nodes forming a set S ⊆ V . The smallest eigenvalue λ(S) of L(S) plays a pivotal role in various dynamics defined on G. For example, λ(S) characterizes the convergence rate of leader-follower consensus, as well as the effectiveness of a pinning scheme for the pinning control problem, with larger λ(S) corresponding to smaller convergence time or better effectiveness of a pinning scheme. In this paper, we focus on the problem of optimally selecting a subset S of fixed k ≪ n nodes, in order to maximize the smallest eigenvalue λ(S) of the grounded Laplacian matrix L(S). We show that this optimization problem is NP-hard and that the objective function is non-submodular but monotone. Due to the difficulty to obtain the optimal solution, we first propose a naïve heuristic algorithm selecting one optimal node at each time for k iterations. Then we propose a fast heuristic scalable algorithm to approximately solve this problem, using derivative matrix, matrix perturbations, and Laplacian solvers as tools. Our naïve heuristic algorithm takes Õ(knm) time, while the fast greedy heuristic has a nearly linear time complexity of Õ(km), where Õ(•) notation suppresses the poly(log n) factors. We also conduct numerous experiments on different networks sized up to one million nodes, demonstrating the superiority of our algorithm in terms of efficiency and effectiveness compared to baseline methods.

show abstract

Maximizing the Number of Spanning Trees in a Connected Graph

Cited by 23 publications

References 54 publications

Reliable Graphs for SLAM

Reliable Graphs for SLAM

Maximizing Influence of Leaders in Social Networks

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Contact Info

Product

Resources

About