Real Time Asymptotic

1 Abstract A random greedy algorithm, somewhat modiied, is analyzed by using a real time context and showing that the variables remain close to the solution of a natural diierential equation. Given a (k + 1)-uniform simple hypergraph on N vertices, regular of degree D, the algorithm gives a packing of disjoint hyperedges containing all but O(ND ?1=k ln c D) of the vertices. Let H = (V; E) be a (k + 1)-uniform hypergraph on N vertices. A packing P is a family of disjoint edges. Given P we correspond the set S = V ? S P of those vertices v not in the packing, these v we call surviving vertices. We shall assume: H is simple. That is, any two vertices are in at most one edge. H is regular of degree D. That is, every vertex v lies in precisely D e 2 E. We are interested in the asymptotics for k xed, D; N ! 1. We assume k 2 is xed throughout. We show Theorem. There exists a packing with jSj = O(ND ?1=k ln c D) where c depends on k. (We make no attempt to optimize c.) Our approach is to give a real time random process that produces a packing with EjSj] meeting these bounds. The process, as described in x1,2, can be thought of as the random greedy algorithm with some \stabilization mechanisms" added. Placing the algorithm in a real time context allows for simulation of the variables by a diierential equation and the analysis of our discrete, albeit asymptotic, procedure becomes quite continuous in nature. The study of asymptotic packing can be said to date from the proof by V. RR odl 3] of a classic conjecture of Paul Erd} os and Haim Hanani 2]. RR odl showed that for l < k xed and n ! 1 there exists a \packing" P of ? n l = ? k l k-element subsets of an n-element universe so that every l points of lie in at most one of the k-sets. This was nicely generalized by N. Pippenger in work appearing 5] jointly with this author. He showed that any k-uniform hypergraph on N vertices with deg(v) D for every v and any two vertices v; w having o(D) common edges has a packing P with jSj = o(n). (Here k is xed, N; D ! 1.) Recent work …