Analysis of properties on the connectivity of graphs and networks and its applications to design of algorithms

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Many problems in communication networks can be modeled as graph problems if we fix several factors describing the problems. In this work, we first modeled problems in communication networks as graph (network) problems by selecting the essential conditions from a lot of complex conditions inquired for the communication problems and then analyzed properties on the connectivity by observing the structures of the problems. Furthermore, we designed approximation algorithms using the mathematical structures characterizing the properties that we elucidated. From our work, many results related to properties on the connectivity of graphs and networks were obtained. In what follows, we classify the results roughly into the multicast tree problem, the queue layout problem, and the minimum k-way cut problem, and then state each summary.1. The multicast tree problem: In this work, we proposed a (3/2+(4/3) ρ)-approximation algorithm for this problem, where ρ is the best achievable approximation ratio for the Steiner tree problem. Compared with the best approximation ratio (2+ρ) in the previously known approximation algorithms, the more ρ is improved, the more the approximation ratio of our algorithm is improved. 2. The queue layout problem. In this work, we studied on the class of iterated line digraphs which contains several digraph classes that were studied individually so far, and present upper and lower bounds on the queuenumber of iterated line digraphs. 3. The minimum k-way cut problem. In this work, we defined a new problem, and designed a 2-approximation algorithm for the minimum k-way cut problem based on a method for the optimum solution for the new problem. Using this result, we improved the time complexity of a 2-approximation algorithm from O(k(mn+n^2logn)) to O(mn+n^2logn).