In the 2-Vertex-Connected Spanning Subgraph problem (2-VCSS), we are given an undirected graph G, and the objective is to find a 2-vertex-connected spanning subgraph S of G with the minimum number of edges. In the context of survivable network design, 2-VCSS is one of the most fundamental and well-studied problems. There has been active research on improving the approximation ratio of algorithms, and the current best ratio is 4/3, achieved by Bosch-Calvo, Grandoni, and Jabal Ameli. In this paper, we improve the approximation ratio to 95/72+ε (< 1.32). The key idea in our algorithm is to introduce a 2-edge-cover without certain cycle components, and use it as an initial solution.

The $2$-Edge-Connected Spanning Subgraph problem (2-ECSS) is one of the most fundamental and well-studied problems in the context of network design. In the problem, we are given an undirected graph $G$, and the objective is to find a $2$-edge-connected spanning subgraph $H$ of $G$ with the minimum number of edges. For this problem, a lot of approximation algorithms have been proposed in the literature. In particular, very recently, Garg, Grandoni, and Ameli gave an approximation algorithm for 2-ECSS with factor $1.326$, which was the best approximation ratio. In this paper, we give a $(1.3+\varepsilon)$-approximation algorithm for 2-ECSS, where $\varepsilon$ is an arbitrary positive fixed constant, which improves the previously known best approximation ratio. In our algorithm, we compute a minimum triangle-free $2$-edge-cover in $G$ with the aid of the algorithm for finding a maximum triangle-free $2$-matching given by Hartvigsen. Then, with the obtained triangle-free $2$-edge-cover, we apply the arguments by Garg, Grandoni, and Ameli.

A general scheme to represent the relation between dynamic images and camera motion is presented, and its application to visual servoing proposed. For a specific object, the camera cannot obtain any arbitrary image, so that the possible combination of the camera pose and the obtained image should be constrained on a lower dimensional hyper surface within the product space of all the combinations. The visual servoing, for example, is interpreted as to find a path on this surface leading to a given goal image. Our approach is to analyse the properties of this surface and utilise its tangential property for visual servoing. We propose to use the principal component analysis and to represent images with a composition of small number of "eigen-images" by using the Karhunen-Loeve expansion. We describe that a normal vector of this surface is related to the so-called interaction matrix. We then present a dynamic estimation of the normal vectors to move the robot arm mounting a camera to a goal position where a given goal image will be obtained. Experimental results of visual servoing method show the feasibility and applicability of our proposed approach.