Abstract.
A truncated permutation matrix polytope is defined as the convex hull of a proper subset of n-permutations represented as 0/1 matrices. We present a linear system that models the coNP-complete non-Hamilton tour decision problem based upon constructing the convex hull of a set of truncated permutation matrix polytopes. Define polytope Pn−1 as the convex hull of all n-1 by n-1 permutation matrices. Each extreme point of Pn−1 is placed in correspondence (a bijection) with each Hamilton tour of a complete directed graph on n vertices. Given any n vertex graph G n , a polynomial sized linear system F(n) is constructed for which the image of its solution set, under an orthogonal projection, is the convex hull of the complete set of extrema of a subset of truncated permutation matrix polytopes, where each extreme point is in correspondence with each Hamilton tour not in G n . The non-Hamilton tour decision problem is modeled by F(n) such that G n is non-Hamiltonian if and only if, under an orthogonal projection, the image of the solution set of F(n) is Pn−1. The decision problem ‘Is the projection of the solution set of F(n)=Pn−1?’ is therefore coNP-complete, and this particular model of the non-Hamilton tour problem appears to be new.
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Dedicated to the 250+ families in Kelowna BC, who lost their homes due to forest fires in 2003.
I visited Ted at his home in Kelowna during this time - his family opened their home to evacuees and we shared happy and sad times with many wonderful people.
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Gismondi, S., Swart, E. A model of the coNP-complete non-Hamilton tour decision problem for directed graphs. Math. Program., Ser. A 100, 471–483 (2004). https://doi.org/10.1007/s10107-003-0480-9
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DOI: https://doi.org/10.1007/s10107-003-0480-9