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We propose a novel way to consider the max-min fairness (MMF) paradigm in traffic engineering. Since MMF appears as a reference model for a fair capacity allocation when the traffic flows are elastic and rates are adapted based on resource availability, we consider it as a requirement due to the way resources are shared by the distributed rate control scheme (like that of the transport protocol), rather than the routing objective. In particular, we define the traffic engineering problem where, given a network topology with link capacities and a set of elastic traffic demands to route, we must select a single path for each demand so as to maximize a network utility function, assuming an MMF bandwidth allocation. We propose a compact mixed-integer linear programming formulation as well as a restricted path formulation. We show with computational experiments that the exact formulation can be solved in a reasonable amount of computing time for medium-size networks and that the restricted path model provides solutions of comparable quality much faster. [ABSTRACT FROM PUBLISHER]