Quasi-P versus P

This report studies problems coming from three di erent domains of theoretical computer science: Clique versus Independent set in communication complexity, the Alon-SaksSeymour conjecture in graph coloring and the stubborn problem in constraint satisfaction problem. Each problem admits a quasi-polynomial size solution, meaning of size n where n is the parameter, and the problem is to nd a polynomial size solution. We prove an equivalence theorem between the existence of a polynomial solution for every problem. We study further the Clique versus Independent set problem on two particular classes of graphs: random graphs and split-free graphs, for which we nd a polynomial size solution.