Learning-Based Heuristic for Combinatorial Optimization of the Minimum Dominating Set Problem using Graph Convolutional Networks

A dominating set of a graph $\mathcal{G=(V, E)}$ is a subset of vertices $S\subseteq\mathcal{V}$ such that every vertex $v\in \mathcal{V} \setminus S$ outside the dominating set is adjacent to a vertex $u\in S$ within the set. The minimum dominating set problem seeks to find a dominating set of minimum cardinality and is a well-established NP-hard combinatorial optimization problem. We propose a novel learning-based heuristic approach to compute solutions for the minimum dominating set problem using graph convolutional networks. We conduct an extensive experimental evaluation of the proposed method on a combination of randomly generated graphs and real-world graph datasets. Our results indicate that the proposed learning-based approach can outperform a classical greedy approximation algorithm. Furthermore, we demonstrate the generalization capability of the graph convolutional network across datasets and its ability to scale to graphs of higher order than those on which it was trained. Finally, we utilize the proposed learning-based heuristic in an iterative greedy algorithm, achieving state-of-the-art performance in the computation of dominating sets

Affine-Invariant Outlier Detection and Data Visualization

A wealth of data is generated daily by social media websites that is an essential component of the Big Data Revolution. In many cases, the data is anonymized before being disseminated for research and analysis. This anonymization process distorts the data so that some essential characteristics are lost which may not be captured by methods that are not robust against such transformations. In this paper we propose novel algorithms, for two-dimensional data, for a recently discovered statistical data analysis measure, the Ray Shooting Depth (RSD) that provides an affineinvariant ranking of data points. In addition, we prove some complexity results and illustrate some of the desirable properties of RSD via comparisons with other similar notions. We develop an open-source data visualization tool based on RSD, and show its applications in distribution estimation, outlier detection, and 2D tolerance-region construction

Controllability Backbone in Networks

This paper studies the controllability backbone problem in dynamical networks defined over graphs. The main idea of the controllability backbone is to identify a small subset of edges in a given network such that any subnetwork containing those edges/links has at least the same network controllability as the original network while assuming the same set of input/leader vertices. We consider the strong structural controllability (SSC) in our work, which is useful but computationally challenging. Thus, we utilize two lower bounds on the network's SSC based on the zero forcing notion and graph distances. We provide algorithms to compute controllability backbones while preserving these lower bounds. We thoroughly analyze the proposed algorithms and compute the number of edges in the controllability backbones. Finally, we compare and numerically evaluate our methods on random graphs.Comment: Accepted in 62nd IEEE Conference on Decision and Control, Dec. 13-15, 2023, Singapor