dc.contributor.author Roh, Patrick dc.date.accessioned 2007-01-19 15:56:56 (GMT) dc.date.available 2007-01-19 15:56:56 (GMT) dc.date.issued 2007-01-19T15:56:56Z dc.date.submitted 2007 dc.identifier.uri http://hdl.handle.net/10012/2658 dc.description.abstract This thesis will address several problems in discrete optimization. These problems are considered hard to solve. However, good approximation algorithms for these problems may be helpful in approximating problems in computational biology and computer science. en Given an undirected graph G=(V,E) and a family of subsets of vertices S, the minimum crossing spanning tree is a spanning tree where the maximum number of edges crossing any single set in S is minimized, where an edge crosses a set if it has exactly one endpoint in the set. This thesis will present two algorithms for special cases of minimum crossing spanning trees. The first algorithm is for the case where the sets of S are pairwise disjoint. It gives a spanning tree with the maximum crossing of a set being 2OPT+2, where OPT is the maximum crossing for a minimum crossing spanning tree. The second algorithm is for the case where the sets of S form a laminar family. Let b_i be a bound for each S_i in S. If there exists a spanning tree where each set S_i is crossed at most b_i times, the algorithm finds a spanning tree where each set S_i is crossed O(b_i log n) times. From this algorithm, one can get a spanning tree with maximum crossing O(OPT log n). Given an undirected graph G=(V,E), and a family of subsets of vertices S, the minimum crossing perfect matching is a perfect matching where the maximum number of edges crossing any set in S is minimized. A proof will be presented showing that finding a minimum crossing perfect matching is NP-hard, even when the graph is bipartite and the sets of S are pairwise disjoint. dc.format.extent 370821 bytes dc.format.mimetype application/pdf dc.language.iso en en dc.publisher University of Waterloo en dc.subject Approximation Algorithms en dc.subject Discrete Optimization en dc.subject Graph Theory en dc.subject NP en dc.title Minimum Crossing Problems on Graphs en dc.type Master Thesis en dc.pending false en dc.subject.program Combinatorics and Optimization en uws-etd.degree.department Combinatorics and Optimization en uws-etd.degree Master of Mathematics en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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