We propose a new approach consisting in combining genetic algorithms and regret-based incremental preference elicitation for solving multi-objective combinatorial optimization problems with unknown preferences. For the purpose of elicitation, we assume that the decision maker's preferences can be represented by a parameterized scalarizing function but the parameters are initially not known. Instead, the parameter imprecision is progressively reduced by asking preference queries to the decision maker during the search to help identify the best solutions within a population. Our algorithm, called RIGA, can be applied to any multi-objective combinatorial optimization problem provided that the scalarizing function is linear in its parameters and that a (near-)optimal solution can be efficiently determined when preferences are known. Moreover, RIGA runs in polynomial time while asking no more than a polynomial number of queries. For the multi-objective traveling salesman problem, we provide numerical results showing its practical efficiency in terms of number of queries, computation time and gap to optimality.