The (r/Xp)-medianoid and (r/p)-centroid problems are two well-known competitive location problems on networks. In the medianoid problem, a player locates r facilities in a network in order to maximize its market share, given that p facilities are already located. In the centroid problem, two players in turn locate p and r facilities at the vertices of a network in order to maximize their market share. The problem is for the first player to find a set of locations that will minimize the second player's maximal market share. These problems are both NP-hard. They are solved by means of tabu search heuristics. Computational results confirm the value of the proposed approach. (A)