We introduce nonlinear formulations for the multiway cut and multicut problems. By simple linearizations of these formulations we derive several well known formulations and valid inequalities as well as several new ones. Through these formulations we establish a connection between the multiway cut and the maximum weighted independent set problem that leads to the study of the tightness of several LP formulations for the multiway cut problem through the theory of perfect graphs. We finally introduce a new randomized rounding heuristic to study the worst case bound of these formulations, obtaining a new bound of 2α(H)(1-1/k) for the multicut problem, where α(H) is the size of a maximum independent set in the demand graph H.