Finding a maximum matching in a sparse random graph in O ( n ) expected time

Finding a Maximum Independent Set in a Sparse Random Graph

We consider the problem of finding a maximum independent set in a random graph. The random graphG is modelled as follows. Every edge is included independently with probability d n , where d is some sufficiently large constant. Thereafter, for some constant α, a subset I of αn vertices is chosen at random, and all edges within this subset are removed. In this model, the planted independent set I...

Finding One Community in a Sparse Graph

We consider a random sparse graph with bounded average degree, in which a subset of vertices has higher connectivity than the background. In particular, the average degree inside this subset of vertices is larger than outside (but still bounded). Given a realization of such graph, we aim at identifying the hidden subset of vertices. This can be regarded as a model for the problem of finding a t...

Coloring Sparse Random k-Colorable Graphs in Polynomial Expected Time

Feige and Kilian [5] showed that finding reasonable approximative solutions to the coloring problem on graphs is hard. This motivates the quest for algorithms that either solve the problem in most but not all cases, but are of polynomial time complexity, or that give a correct solution on all input graphs while guaranteeing a polynomial running time on average only. An algorithm of the first ki...

The Maximum Number of Edges in a Strongly Multiplicative Graph

The maximum order of a strong matching in a random graph

A strong matching S in a given graph G is a set of disjoint edges {el' e2, ... , em} such that no other edge of the graph G connects an end-vertex of ei with an end-vertex of ej,(ei =Iej). Let Gn,p be the random graph on n vertices with fixed edge probability p, 0 < p < 1. It is shown that, with probability tending to 1 as n ~ 00, the maximum size f3 of a strong matching in Gn,p satisfies where...