Author
Abstract
For a connected graph $$\mathcal {G}=(V,E)$$ G = ( V , E ) with n nodes, m edges, and Laplacian matrix $${\varvec{ L }}$$ L , a grounded Laplacian matrix $${\varvec{ L }}(S)$$ L ( S ) of $$\mathcal {G}$$ G is a $$(n-k) \times (n-k)$$ ( n - k ) × ( n - k ) principal submatrix of $${\varvec{ L }}$$ L , obtained from $${\varvec{ L }}$$ L by deleting k rows and columns corresponding to k selected nodes forming a set $$S\subseteq V$$ S ⊆ V . The smallest eigenvalue $$\lambda (S)$$ λ ( S ) of $${\varvec{ L }}(S)$$ L ( S ) plays a pivotal role in various dynamics defined on $$\mathcal {G}$$ G . For example, $$\lambda (S)$$ 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 $$\lambda (S)$$ 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 $$k \ll n$$ k ≪ n nodes, in order to maximize the smallest eigenvalue $$\lambda (S)$$ λ ( S ) of the grounded Laplacian matrix $${\varvec{ L }}(S)$$ 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 of obtaining 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 solve this problem, using the derivative matrix, matrix perturbations, and Laplacian solvers as tools. Our naïve heuristic algorithm takes $$\tilde{O}(knm)$$ O ~ ( k n m ) time, while the fast greedy heuristic has a nearly linear time complexity of $$\tilde{O}(km)$$ O ~ ( k m ) , where $$\tilde{O}(\cdot )$$ O ~ ( · ) notation suppresses the $$\textrm{poly} (\log n)$$ 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.
Suggested Citation
Xiaotian Zhou & Run Wang & Wei Li & Zhongzhi Zhang, 2025.
"Maximizing the smallest eigenvalue of grounded Laplacian matrix,"
Journal of Global Optimization, Springer, vol. 91(4), pages 807-828, April.
Handle:
RePEc:spr:jglopt:v:91:y:2025:i:4:d:10.1007_s10898-025-01470-z
DOI: 10.1007/s10898-025-01470-z
Download full text from publisher
As the access to this document is restricted, you may want to search for a different version of it.
Corrections
All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:91:y:2025:i:4:d:10.1007_s10898-025-01470-z. See general information about how to correct material in RePEc.
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
We have no bibliographic references for this item. You can help adding them by using this form .
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .
Please note that corrections may take a couple of weeks to filter through
the various RePEc services.