On Linear Characterizations of Combinatorial Optimization Problems

Richard M. Karp, Christos H. Papadimitriou

Abstract: We show that there can be no computationally tractable description by linear inequalities of the polyhedron associated with any NP-complete combinatorial optimization problem unless NP = co-NP -- a very unlikely event. We also apply the ellipsoid method for linear programming to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities.