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On the circuit diameter of some combinatorial polytopes. (English) Zbl 1416.52006
The combinatorial diameter of a polytope \(P\) on \(n\) vertices is the maximum value of a shortest path in the \(1\)-skeleton of \(P\) between two vertices in \(P\). The combinatorial diameter is linked to longstanding open questions about the existence of a pivoting rule that yields a polynomial running time for the Simplex Method. The circuit diameter of a polytope \(P\) is the maximum value of a shortest path in the \(1\)-skeleton of \(P\) between two vertices of \(P\) that uses potential edge direction of \(P\), that is if \(P = \{\tilde{x}\in {\mathbb{R}}^n : A\tilde{x} = \tilde{b}, \ \ B\tilde{x}\leq \tilde{d}\}\) where \(A\) and \(B\) are rational matrices, and \(\tilde{b}\) and \(\tilde{d}\) are rational vectors, then the circuits of \(P\) are the set of potential edge directions that can arise by varying \(\tilde{b}\) and \(\tilde{d}\).
In this paper, the authors study the circuit diameter of the matching polytope, the perfect matching polytope, the Traveling Salesman (TSP) polytope, and the fractional stable set polytope. An exact characterization of the circuit diameter of the matching polytope is given, in particular is is shown to be equal to \(2\) for \(n\geq 7\). The circuit diameter of the perfect matching polytope on \(n\) vertices is shown to be equal to \(1\) for \(n\neq 8\) and equal to \(2\) for \(n=8\). An exact characterization of the circuit diameter of the TSP polytope is given. It is shown to be equal to \(1\) for \(n\neq 5\) and equal to \(2\) for \(n=5\). Finally, a complete characterization of the circuit diameter of the fractional stable set polytope is presented. It is shown to be essentially upper bounded by the diameter of \(P\), which is often significantly smaller than \(n\), the number of vertices of \(P\).

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
Full Text: DOI
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