Tropical polar cones, hypergraph transversals, and mean payoff games

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• 作者：Xavier Allamigeon^a ; ^{xavier.allamigeon@inria.fr"" rel=""nofollow} ; Sté ; phane Gaubert^a ; ^{stephane.gaubert@inria.fr"" rel=""nofollow} ; Ricardo D. Katz^b ; ^{rkatz@fceia.unr.edu.ar"" rel=""nofollow}
• 关键词：Max-plus semiring ; Max-plus convexity ; Tropical convexity ; Polyhedra ; Hypergraph transversals ; Minimal hitting sets ; Minimal solutions
• 刊名：Linear Algebra and its Applications
• 出版年：2011
• 出版时间：1 October 2011
• 年：2011
• 卷：435
• 期：7
• 页码：1549-1574
• 全文大小：546 K

文摘

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.