Abstract

We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of relations =, , ? and 6=. Various constrained versions of Max FLS, where a subset of relations must be satisfied or where the variables take bounded discrete values, are also considered. We establish the complexity of solving these problems optimally and, whenever they are intractable, we determine their degree of approximability. Max FLS with =, or ? relations is NP-hard even when restricted to homogeneous systems with bipolar coefficients, whereas it can be solved in polynomial time for 6= relations with real coefficients. The various NP-hard versions of Max FLS belong to different approximability classes depending on the type of relations and the additional constraints. We show that the ran...

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