| Research Abstract |
In this research, we have developed efficient graph algorithms and clarified graph structures in the graph/network problems.
We have obtained the following results for the problems related to connectivity. We have improved the time complexity for representing all minimum cuts in a cactus. To achieve this, we used a maximum adjacency ordering, a graph search procedure by which all minimum cuts can be found without using the maximum-flow algorithm. A subset of edges is called a k-cut if removal of it results in k components. We have reduced the time bound for computing a minimum k-cut for k=3,4,5,6 by a new approach that enumerates 2-cut in the nondecreasing order of weights. We have proved that necessary information to solve the k-edge-connectivity augmentation problem can be extracted from an appropriate set of cuts with size less than k in a given graph. By using such a set of cuts, we can control structure of solutions to the edge-connectivity augmentation problem.
We have also obtained the following results in designing approximation algorithms. For the vertex-connectivity problem with a target value k, an approximation algorithm was proposed only for the case where a given graph is (k-1)-vertex-connected. We extend the algorithm so that it works for an arbitrary input graph. Our algorithm delivers an solution with absolute error 2(α-k)k, where α =the vertex-connectivity of an input graph. We have studied the problem of increasing the edge- and vertex-connectivities at the same time, and gave an approximation algorithm with absolute error that depends only on the target values. We have also designed a (7/2)-approximation algorithm for the weighted 3-vertex-connectivity augmentation problem, a 2-approximation algorithm for the weighted minimum edge dominating set problem and a dH(r)-approximation algorithm for the network design problem in hypergraph with degree d, where r is the maximum demand and H() is the harmonic function.
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