A multicast game is a network design game modelling how selfish non-cooperative agents build and maintain one-to-many network communication. There is a special source node and a collection of agents located at corresponding terminals. Each agent is interested in selecting a route from the special source to its terminal minimizing the cost. The mutual influence of the agents is determined by a cost sharing mechanism, which evenly splits the cost of an edge among all the agents using it for routing. In this paper we provide several algorithmic and complexity results on finding a Nash equilibrium minimizing the value of Rosenthal potential. Let n be the number of agents and G be the communication network. We show that for a given strategy profile s and integer k ≥ 0, there is a local search algorithm which in time n O(k)⋅|G| O(1) finds a better strategy profile, if there is any, in a k-exchange neighbourhood of s. In other words, the algorithm decides if Rosenthal potential can be decreased by changing strategies of at most k agents. The running time of our local search algorithm is essentially tight: unless F P T = W, for any function f(k), searching of the k-neighbourhood cannot be done in time f(k)⋅|G| O(1). We also show that an equilibrium with minimum potential can be found in 3 n ⋅|G| O(1) time.