Largest planar matching in random bipartite graphs

Given a distribution G over labeled bipartite (multi) graphs, G = (W; M; E) where jWj = jMj = n, let L(n) denote the size of the largest planar matching of G (here W and M are posets drawn on the plane as two ordered rows of nodes, an upper and a lower one, and a (w; m) edge is drawn as a straight line between w and m). The main focus of this work is to understand the asymptotic (in n) behavior of L(n) for diierent distributions G. Two well studied particular instances of this problem are Ulam's longest increasing subsequence problem and the longest common subsequence problem. This work's main focus is in the case where G is the uniform distribution over the k{regular bipartite graphs on W and M. For k = O(n 1=5?"), we establish that L(n)= p kn tends to 2 in probability when n ! 1. When k = O(1) the convergence in mean to the same limit holds. It is also shown that when each of the n 2 possible edges between W and M are chosen independently with probability 0 < p < 1, then L(n)=n tends to a constant p in probability and in mean when n ! 1. The problems addressed in this work can be thought of as a novel generalization of Ulam's longest increasing subsequence problem and the longest common subsequence problem.