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We consider fair allocation of indivisible items under additive utilities.When the utilities can be negative, the existence and complexity of anallocation that satisfies Pareto optimality and proportionality up to one item(PROP1) is an open problem. We show that there exists a stronglypolynomial-time algorithm that always computes an allocation satisfying Paretooptimality and proportionality up to one item even if the utilities are mixedand the agents have asymmetric weights. We point out that the result does nothold if either of Pareto optimality or PROP1 is replaced with slightly strongerconcepts.

A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation