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- Title
- CONTRIBUTIONS TO ALGORITHMIC MATROID PROBLEMS
- Creator
- Huang, Jinyu
- Date
- 2015, 2015-07
- Description
-
In this thesis, we obtain several algorithms for problems related to matroids, a structure that generalizes the concept of linear independence...
Show moreIn this thesis, we obtain several algorithms for problems related to matroids, a structure that generalizes the concept of linear independence in a vector space and an acyclic subgraph structure in a graph. Matroids have been widely applied in combinatorial optimization, graph theory, coding theory and so forth. Specifically, our results include: In this thesis, we present a constant-competitive online algorithm of the matroid secretary problem for the partition matroids without information of the partition and for the paving matroids. We also introduce the multi-objective matroid secretary problem that extends the matroid secretary problem, in which the weight function is a k-vector w = [w1, · · · , wk]. We show a constant competitive algorithm of the multiobjective matroid secretary problem for the uniform matroids and for the paving matroids. Since bases of a matroid generalize many important combinatorial structures, many counting problems can be expressed as a problem that counts the number of bases of a matroid. An efficient approximate counting algorithm can be designed provided that a rapidly-mixing Markov chain that samples bases of a matroid can be constructed. Let Φ(G) be the conductance of the base-exchange graph G. Matroid- Expansion Conjecture (1989, Mihai and Vazirani) states that Φ(G) ≥ 1 for any base-exchange graph G of a matroid, which implies an FPRAS (fully-polynomial randomized approximation scheme) for counting the number of bases of a matroid. We use λ2, the second smallest eigenvalue of L, the discrete Laplacian matrix of G, to prove the Matroid-Expansion Conjecture for any paving matroid, for any balanced matroid, and for the direct sum of a paving matroid with a balanced matroid. A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. Finally, we prove that if there is a black-box NC algorithm for PIT (Polynomial Identity Testing), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).
Ph.D. in Applied Mathematics, July 2015
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