Implementation and evaluation of graph approximation algorithms

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In this project, we have mainly implemented the following three approximation algorithms : From the experiments, we can conclude :・For bandwidth minimization problem we implemented an approximation algorithm based on volume respecting embeddings (VRE) method invented by U. Feige. For general graphs the experimental results of VRE is not better than that of GPS algorithm (Cuthill-Mckee method), however the results of VRE is significantly better than that of GPS algorithm for some special graph classes such as stretched complete bipartite graphs.・For several matroid covering problems we implemented approximation and heuristic algorithms. For cographic matroid covering problem, our proposed algorithm is comparable to a natural greedy algorithm. However for graphic and transversal matroid covering problems, the results of our proposed algorithm is inferior to that of natural greedy algorithms in quality. Our proposed algorithms can be implemented more easily than exact algorithms.・ For maximum weight independent set problem for d-claw free graphs several approximation algorithms are proposed. We implemented some of them ; SizeTwolmp introduced by V.Bafna et. al, Bestlmp proposed by B.Chandra et al., SquareImp developed by P.Berman and several algorithms of greedy type. The experimental results showed that SizeTwoImp, Bestlmp, and Squarelmp can find better solutions than greedy algorithms in reasonable time. The three approximation algorithms are practicable.Some theoretical results were obtained as a by-product through this research : Lower bounds of graph parameters called "vertex boundary-width" and "path distance-width". Vertex boundary-width can be considered as a lower bound of bandwidth and path distance-width is strongly related to approximating the bandwidth. We also obtained other theoretical results such as on unit grid intersection graphs, longest induced path problem for k-chordal graphs and tree-length. These are indirectly related to the research project.