We consider multi-objective combinatorial optimization problems where preferences can be represented by a parameterized scalarizing function that is linear in its parameters. Assuming that the parameters are initially not known, we iteratively collect preference information from the decision maker until being able to identify her preferred solution. To obtain informative preference information, we ask the decision maker to compare two promising solutions generated using the extreme points of the polyhedron representing the admissible parameters. Moreover, our stopping criterion guarantees that the returned solution is optimal (or near-optimal) according to the decision maker's preferences. Finally, we provide numerical results to demonstrate the practical efficiency of our approach.