Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Real Time Asymptotic Packing. | Real Time Asymptotic Packing Joel Spencer spencer@ Courant Institute New York Submitted May 7 1966. Accepted June 12 1996. 1 Abstract A random greedy algorithm somewhat modified is analyzed by using a real time context and showing that the variables remain close to the solution of a natural differential 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 lnc 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 u 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 De 2 E. We are interested in the asymptotics for k fixed D N 1. We assume k 2 is fixed throughout. We show Theorem. There exists a packing with SI O ND-1 k lnc 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 E S meeting these bounds. The process as described in 1 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 di erential 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. Rodl 3 of a classic conjecture of Paul Erdos and Haim Hanani 2 . Rodl showed that for l k fixed and n 1 there exists a packing P of n k 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 .
