Polytopes and Complexity

Polyhedral combinatorics is one of the oldest and most important techniques for the algorithmic solution of combinatorial optimization problems. A host of efficient algorithms for combinatorial optimization problems have a “polyhedral interpretation”, and in fact for many the polyhedral approach was instrumental in the derivation of the algorithm and its proof. In this paper we review some recent results which relate the polyhedral theory to the theory of computational complexity. These results reveal the limitations of this approach in solving NP-complete problems. They focus on the computational complexity of certain elements of the structure (such as the adjacency relation, the facets, the supporting hyperplanes, and the interior) of the polytopes of NP-complete problems in general – and, in some cases, of the traveling salesman problem in particular. The thrust of these results is that most of these elements are extremely complex (NP-complete or worse). Since the understanding of these elements is at the root of “polyhedral” algorithms, these results seem to suggest that the polytopal approach is not likely to yield feasible algorithms for NP-complete combinatorial optimization problems. Finally, we present a class of persistent open problem, pertaining to whether a given hyperplane contains any vertex of a fixed “nice” polytope, such as the matching, or even the assignment polytope.