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.
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Ardekani, L.H., Arthanari, T.S. (2013). The Multi-flow Necessary Condition for Membership in the Pedigree Polytope Is Not Sufficient- A Counterexample. In: Selamat, A., Nguyen, N.T., Haron, H. (eds) Intelligent Information and Database Systems. ACIIDS 2013. Lecture Notes in Computer Science(), vol 7803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36543-0_42
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