Branch-and-Cut for the Maximum Feasible Subsystem Problem

This paper presents a branch-and-cut algorithm for the NPhard maximum feasible subsystem problem: For a given infeasible linear inequality system, determine a feasible subsystem containing as many inequalities as possible. The complementary problem, where one has to remove as few inequalities as possible in order to make the system feasible, can be formulated as a set covering problem. The rows of this formulation correspond to irreducible infeasible subsystems, which can be exponentially many. It turns out that the main issue of a branchand-cut algorithm for the maximum feasible subsystem problem is to efficiently find such infeasible subsystems. We present three heuristics for the corresponding NP-hard separation problem and discuss cutting planes from the literature, such as set covering cuts of Balas and Ng, Gomory cuts, and {0, 1 2 }-cuts. Furthermore, we compare a heuristic of Chinneck and a simple greedy algorithm. The main contribution of this paper is an extensive computational study on a variety of instances arising in a number of applications.