Linear Programming complementation and its application to fractional graph theory
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In this paper, we introduce a new kind of duality for Linear Programming (LP), that we call LP complementation. We prove that the optimal values of an LP and of its complement are in bijection (provided that either the original LP or its complement has an optimal value greater than one). The main consequence of the LP complementation theorem is for hypergraphs. We introduce the complement of a hypergraph and we show that the fractional packing numbers of a hypergraph and of its complement are in bijection; similar results hold for fractional matching, covering and transversal numbers. This hypergraph complementation theorem has several consequences for fractional graph theory. In particular, we relate the fractional dominating number of a graph to the fractional total dominating number of its complement. We also show that the edge toughness of a graph is equal to the fractional transversal number of its cycle matroid. We then consider the following particular problem: let G be a graph and b be a positive integer, then how many vertex covers of G, say S_1, ..., S_t_b, can we construct such that every vertex appears at most b times in total? The integer b can be viewed as a budget we can spend on each vertex, and given this budget we aim to cover all edges for as long as possible (up to time t_b). We then prove that t_b ∼χ_f/χ_f - 1 b, where χ_f is the fractional chromatic number of G.
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