A Memetic Algorithm for the Generalized Minimum Vertex-Biconnected Network Problem Bin Hu Vienna University of Technology Favoritenstraße 9–11/1861 1040 Vienna, Austria hu@ads.tuwien.ac.at G¨ unther R. Raidl Vienna University of Technology Favoritenstraße 9–11/1861 1040 Vienna, Austria raidl@ads.tuwien.ac.at Abstract—The generalized minimum vertex-biconnected net- work problem plays an important role in the design of surviv- able backbone networks that should be fault tolerant to single component outage. When given a graph where the nodes are partitioned into clusters, the goal is to find a subgraph of minimum costs that connects exactly one node from each cluster in a vertex-biconnected way. We present a memetic algorithm that uses fast local improvement methods to produce high quality solutions and an intelligent crossover operator which controls the balance between diversity and intensity in the population. Tests on Euclidean TSPlib instances with up to 442 nodes show that this approach is highly efficient. Index Terms—Network Design; Biconnectivity; Memetic Algo- rithm; I. I NTRODUCTION The Generalized Minimum Vertex-Biconnected Network Problem (GMVBCNP) is defined as follows. We consider a complete, undirected weighted graph G = 〈V,E,c〉 with node set V , edge set E, and edge cost function c : E → R + . Node set V is partitioned into r pairwise disjoint clusters V 1 ,V 2 ,...,V r ,  r i=1 V i = V, V i ∩ V j = ∅ ∀i, j = 1,...,r, i = j . A solution to the GMVBCNP defined on G is a subgraph S = 〈P,T 〉,P = {p 1 ,...,p r }⊆ V connecting exactly one node from each cluster, i.e., p i ∈ V i , ∀i =1,...,r, and the removal of any single node v ∈ P along with all its incident edges would not disconnect S. An example is given in Figure 1. The costs of such an edge-biconnected network are its total edge costs, i.e., c(T )= ∑ (u,v)∈T c(u, v), and the objective is to identify a feasible solution with minimum costs. In practice, when designing large communication networks, devices belonging to the same local area network can be modelled as nodes within the same cluster. A backbone is required to connect one device per local network that can be regarded as its supernode. When additionally requiring fault tolerance by means of edge-biconnectivity, the network is able to survive single connection outages. However, if a supernode that is a so-called cut point fails, the network still breaks into multiple disconnected components. With the vertex-biconnectivity property, this issue is finally covered as well. For approaching the GMVBCNP, we suggest a Memetic Algorithm (MA) [1] that uses problem specific variation opera- V 1 V 2 V 3 V 4 V 5 p 1 p 2 p 3 p 4 p 5 V 6 p 6 Fig. 1. Example for a solution to the GMVBCNP. tors and local improvement procedures in order to enhance the solution quality. MAs can be seen as hybrid techniques that combine the population concept of Evolutionary Algorithms (EA) with intensification mechanisms that exploit the specific knowledge on the actual problem. Therefore MAs show great capabilities in inheriting the benefits of EAs while compen- sating some of their weaknesses in order to find high quality solutions to difficult optimization problems. To the best of our knowledge, the GMVBCNP has not been addressed in the literature yet. However, there are works that study the strongly related Generalized Minimum Edge Bicon- nected Network Problem (GMEBCNP) which only requires edge-biconnectivity. The GMEBCNP has been introduced by Huygens [2]. He proposed integer programming formulations, but no practical results on actual instances were published. Leitner et al. [3], [4] presented Variable Neighborhood Search (VNS) approaches for the GMEBCNP based on several types of neighborhood structures which augment each other well. Some of these neighborhoods use enhanced techniques to reduce the search space. They can be adapted and applied on the current problem as well. When neglecting the generalization, we get the classical problem of finding a minimum-cost vertex-biconnected net- work on a given graph which was introduced by Eswaran et al. [5]. They showed that this problem is NP-hard and introduced the so-called block-cut graph that allows efficient detection of cut point. This method will be used in this work as well.