 Maximum feasible subsystem

Given finitely encoded linear inequalities, a maximum feasible subsystem is a largest collection that can be simultaneously satisfied. Its decision version asks whether some $k$ inequalities admit a common solution. This is NP-complete, despite polynomial time feasibility testing for each fixed collection. For a graph, use nonnegative variables $x_v$, constraints $x_v\ge 1$ for vertices, and $M>|V|$ copies of $x_u+x_v\le 1$ for each edge. A subsystem of size $M|E|+k$ must retain a copy of every edge constraint and at least $k$ vertex constraints; those vertices form an independent set. Conversely an independent set of size $k$ supplies such a feasible subsystem by its indicator vector.