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On the Existence of Ideal Solutions in Multi-objective 0-1 Integer Programs

We study conditions under which the objective functions of a multi-objective 0-1 integer linear program guarantee the existence of an ideal point, meaning the existence of a feasible solution that simultaneously minimizes all objectives. In addition, we study the complexity of recognizing whether a set of objective functions satisfies these conditions: we show that it is NPhard, but can be solved in pseudo-polynomial time. Few multi-objective 0-1 integer programs have objectives satisfying the conditions for the existence of an ideal point, but the conditions may be satisfied for a subset of the objective functions and/or be satisfied when the objective functions are restricted to a subset of the variables. We illustrate how such occurrences can be exploited to reduce the number of objective functions and/or to derive cuts in the space of the decision variables.