Several Aspects of Antimatroids and Convex Geometries Master's Thesis
نویسنده
چکیده
Convexity is important in several elds, and we have some theories on it. In this thesis, we discuss a kind of combinatorial convexity, in particular, antimatroids and convex geometries. An antimatroid is a combinatorial abstraction of convexity. It has some di erent origins; by Dilworth in lattice theory, by Edelman and Jamison in the notions of convexity, by Korte{Lov asz who were motivated by scheduling problems. A convex geometry is known as a dual object of an antimatroid. In this thesis, we have four main topics. The rst topic is a characterization result. We characterize line-search antimatroids of rooted digraphs by their forbiddenminors. It implies that the minor theorem for antimatroids does not hold, while it holds for graphs. The second topic is an antimatroidal analogue of Dilworth's decomposition theorem for partially ordered sets. We show a characterization of coatomic antimatroids via the concept of circuits. The third topic is related to submodular-type optimization, which discusses what the \good combinatorial structure" for optimization is. We consider a submodulartype optimization problem on the extreme sets of a convex geometry. We introduce a new kind of submodularity, called c-submodularity, and show that the equivalence between the validity of a greedy algorithm and the c-submodularity of a given function. Moreover, we give a new characterization of a poset shelling, related to this result. The fourth topic is an application of convex geometries to the theory of cooperative games. We show that, if a game on a convex geometry is quasi-convex, then the core is a unique stable set of the game.
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