The goal is to construct, from the input integers, some m subsets.
This means that they have no fully polynomial-time approximation scheme (FPTAS) unless P=NP.
This can be shown by a reduction from the equal-cardinality partition problem (EPART): The following approximation algorithms are known:[1] The fair subset sum problem[4] (FSSP) is a generalization of SSP in which, after the subset is selected, its items are allocated among two or more agents.
The utility of each agent equals the sum of weights of the items allocated to him/her.
However, there are pseudopolynomial time algorithms for enumerating all Pareto-optimal solutions when there are two agents:[5] Nicosia, Pacifici and Pferschy study the price of fairness, that is, the ratio between the maximum sum of utilities, and the maximum sum of utilities in a fair solution: In both cases, if the item value is bounded by some constant a, then the POF is bounded by a function of a.