Some structural and algorithmic properties of the maximum feasible subsystem problem

We consider the problem MAX FS: For a given infeasible linear system, determine a largest feasible subsystem. This problem has interesting applications in linear programming as well as in fields such as machine learning and statistical discriminant analysis. MAX FS is NP-hard and also difficult to approximate, fn this paper we examine structural and algorithmic properties of MAX FS and of irreducible infeasible subsystems (IISs), which are intrinsically related, since one must delete at least one constraint from each IIS to attain feasibility. In particular, we establish: (i) that finding a smallest cardinality IIS is NP-hard as well as very difficult to approximate; (ii) a new simplex decomposition characterization of IISs; (iii) that for a given clutter, realizability as the IIS family for an infeasible linear system subsumes the Steinitz problem for polytopes; (iv) some results on the feasible subsystem polytope whose vertices are incidence vectors of feasible subsystems of a given infeasible system.