The multi-flow necessary condition for membership in the pedigree polytope is not sufficient - A counterexample

Abstract

The multistage insertion formulation (MI) for the symmetric traveling salesman problem (STSP), gives rise to a combinatorial object called pedigree. Pedigrees are in one-to-one correspondence with Hamiltonian cycles. The convex hull of all the pedigrees of a problem instance is called the pedigree polytope. The MI polytope is as tight as the subtour elimination polytope when projected into its two-subscripted variable space. It is known that the complexity of solving a linear optimization problem over a polytope is polynomial if the membership problem of the polytope can be solved in polynomial time. Hence the study of membership problem of the pedigree polytope is important. A polynomially checkable necessary condition is given by Arthanari in [5]. This paper provides a counter example that shows the necessary condition is not sufficient. © 2013 Springer-Verlag.